Ivan Babenko
email: ivan.babenko@umontpellier.fr

St Éphane Sabourau
email: stephane.sabourau@u-pec.fr

VOLUME ENTROPY SEMI-NORM AND SYSTOLIC VOLUME SEMI-NORM

Keywords: 2020 Mathematics Subject Classification. Primary 53C23; Secondary 57N65 Volume entropy, simplicial volume, functorial geometric semi-norms, systolic volume

We introduce the volume entropy semi-norm and the systolic volume semi-norm in real homology and show that they satisfy functorial properties similar to the ones of the simplicial volume. Along the way, we also establish a roughly optimal upper bound on the systolic volume of the multiples of any homology class. Finally, we prove that the volume entropy semi-norm, the systolic volume semi-norm and the simplicial volume semi-norm are equivalent in every dimension.

relation with curvature (e.g., sectional, Ricci, scalar) properties for instance. Yet, the corresponding invariants have not been properly defined or studied as semi-norms, except for the simplicial volume which is a purely topological invariant. (General comparison results between functorial topological semi-norms in relation with the simplicial volume semi-norm have recently been established in [START_REF] Crowley | Functorial seminorms on singular homology and (in)flexible manifolds[END_REF] and [START_REF] Fauser | Exotic finite functorial semi-norms on singular homology[END_REF].) As part of this program to investigate the interactions between geometry and topology through the study of functorial geometric semi-norms, we introduce the volume entropy semi-norm and the systolic volume semi-norm in real homology and carry out a systematic study of these invariants. Both semi-norms require a substantial amount of work in order to properly define them. The volume entropy semi-norm relies on the notion of volume entropy, a geometric invariant of considerable interest closely related to the dynamics of the geodesic flow and the growth of the fundamental groups. The systolic volume semi-norm rests on a new asymptotically optimal estimate in systolic geometry. Both the volume entropy semi-norm and the systolic volume semi-norm share similar functorial properties with the simplicial volume semi-norm (also called the Gromov semi-norm). The equivalence of the three semi-norms in real homology is established in this article.

Let M be a connected closed m-dimensional manifold with a Riemannian metric g. Let H ¡ π 1 (M ) be a normal subgroup of the fundamental group of M . The volume entropy (or simply entropy) of (M, g) relative to H, denoted by ent H (M, g), is the exponential growth rate of the volume of balls in the Riemannian covering M H corresponding to the normal subgroup H ¡ π 1 (M ), that is, π 1 (M H ) = H. More precisely, it is defined as

ent H (M, g) = lim R→∞ 1 R log[vol B H (R)] (1.1)
where B H (R) is a ball of radius R centered at any point in the covering M H . The limit exists and does not depend on the center of the ball. When H is trivial, the covering M H is the universal covering M of M and we simply denote its volume entropy by ent(M, g) without any reference to H. Note that ent H (M, g) ≤ ent(M, g)

for every normal subgroup H ¡ π 1 (M ). The definition extends to connected closed pseudomanifolds, see Definition 2.1, to connected finite graphs and more generally, to finite simplicial complexes with a length metric.

The importance of this notion was first noticed by Efremovich [START_REF] Efremovich | On proximity geometry of Riemannian manifolds[END_REF]. Subsequently, Švarc [START_REF] Švarc | A volume invariant of coverings[END_REF] and Milnor [START_REF] Milnor | A note on curvature and fundamental group[END_REF] related the growth of the volume of balls in the universal covering M to the growth of the fundamental group π 1 (M ) of M . Note that the volume entropy of a connected closed Riemannian manifold is positive if and only if its fundamental group has exponential growth. The connection with the dynamics of the geodesic flow was established by Dinaburg [START_REF] Dinaburg | A connection between various entropy characterizations of dynamical systems[END_REF] and Manning [START_REF] Manning | Topological entropy for geodesic flows[END_REF]. More specifically, the volume entropy bounds from below the topological entropy of the geodesic flow on a connected closed Riemannian manifold and the two invariants coincide when the manifold is nonpositively curved; see [START_REF] Manning | Topological entropy for geodesic flows[END_REF].

The minimal volume entropy of a closed m-pseudomanifold M relative to a normal subgroup H ¡ π 1 (M ) is defined as

ω H (M ) = inf g ent H (M, g) vol(M, g) 1 m
where g runs over the space of all piecewise Riemannian metrics on M . For convenience, we also introduce Ω H (M, g) = ent H (M, g) m vol(M, g)

and Ω H (M ) = inf g ent H (M, g) m vol(M, g)

where g runs over the space of all piecewise Riemannian metrics on M . As previously, if H is trivial, we drop the subscript H.

As an example, the minimal volume entropy of a closed m-manifold M which carries a hyperbolic metric is attained by the hyperbolic metric and is equal to (m-1) vol(M, hyp) 1 m ; see [START_REF] Katok | Entropy and closed geodesics[END_REF], [START_REF] Besson | Volume et entropie minimale des espaces localement symétriques[END_REF] for m = 2 and [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] for m ≥ 3. Furthermore, the minimal volume entropy of a closed manifold which carries a negatively curved metric is positive; see [START_REF] Gromov | Volume and bounded cohomology[END_REF].

For a connected closed orientable m-manifold M , the minimal volume entropy of M is a homotopy invariant, see [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF], which only depends on the image h * ([M ]) ∈ H m (π 1 (M ); Z) of the fundamental class of M by the homomorphism induced by the classifying map h : M → K(π 1 (M ), 1) of M in homology; see [START_REF] Brunnbauer | Homological invariance for asymptotic invariants and systolic inequalities[END_REF].

This homological invariance leads us to consider the volume entropy of a homology class as follows. Given a path-connected topological space X, the volume entropy of a homology class a ∈ H m (X; Z) is defined as ω(a) = inf (M,f )

ω ker f * (M ) (1.2)
where the infimum is taken over all m-dimensional geometric cycles (M, f ) representing a, that is, over all maps f : M → X from an oriented connected closed m-pseudomanifold M to X such that f * ([M ]) = a. As previously, we define Ω(a) = ω(a) m .

For every map f : X → Y between two path-connected topological spaces and every a ∈ H m (X; Z), we have Ω(f * (a)) ≤ Ω(a).

(1.3) By [START_REF] Brunnbauer | Homological invariance for asymptotic invariants and systolic inequalities[END_REF]Theorem 10.2], every orientable connected closed m-manifold M with m ≥ 3 satisfies Ω(M ) = Ω(h * ([M ])) (1.4) where h : M → K(π 1 (M ), 1) is the classifying map of M .

The following result shows that Ω induces a pseudo-distance in homology.

Theorem 1.1. Let X be a path-connected topological space. Then for every a, b ∈ H m (X; Z), we have Ω(a + b) ≤ Ω(a) + Ω(b). In particular, the quantity Ω(a -b) defines a pseudo-distance between a and b in H m (X; Z), Thus, for every homology class a ∈ H m (X; Z), the sequence Ω(k a) is sub-additive. As a result, we can apply the following stabilization process and define

a E = lim k→∞ Ω(k a) k . (1.5)
Note that • E is homogenous, that is, k a E = |k| a E for every k ∈ Z. By homogeneity and density of H m (X; Q) in H m (X; R), this functional extends to a functional on H m (X; R), still denoted by • E .

For an orientable connected closed m-manifold M , define

M E = [M ] E where [M ] ∈ H m (M ; Z) is the fundamental class of M .
The following result, which is a direct consequence of Theorem 1.1, justifies the use of the term volume entropy semi-norm to designate the functional • E . Corollary 1.2. Let X be a path-connected topological space. Then the functional • E is a semi-norm on H m (X; R).

Functorial properties of the volume entropy semi-norm are described in Section 3, where it is shown that the volume entropy semi-norm of a closed orientable manifold depends only on the image of its fundamental class under the classifying map.

The simplicial volume is a much-studied topological invariant sharing similar properties with the volume entropy semi-norm. Let us recall its definition and its basic properties, referring to [START_REF] Gromov | Volume and bounded cohomology[END_REF] for foundational constructions and results regarding this invariant. Let X be a pathconnected topological space. Every real singular m-chain c ∈ C m (X; R) of X is a real linear combination of singular simplices f s : ∆ m → X, that is, c = s r s f s where r s ∈ R. The 1 -norm on the real chain complex is defined as

c 1 = s |r s |.
The simplicial volume of a real homology class a ∈ H m (X; R) is defined as

a ∆ = inf c c 1
where the infimum is taken over all real singular m-cycles c representing a. The simplicial volume of an integral homology class is defined as the simplicial volume of the corresponding real homology class. It is clear that the simplicial volume • ∆ is a functorial semi-norm on H m (X; R). This means that the real homology of a topological space with its simplicial volume semi-norm defines a functor from the category of topological spaces (whose morphisms are continuous maps) to the category of semi-normed vector spaces (whose morphisms are seminorm-nonincreasing homomorphisms); see [START_REF] Borceux | Higher order sheaves and Banach modules[END_REF]. In other words, every continuous map between topological spaces induces a semi-norm-nonincreasing homomorphism in real homology.

As previously, for an orientable connected closed m-manifold M , we let

M ∆ = [M ] ∆ where [M ] ∈ H m (M ; Z) is the fundamental class of M . By [24, §3.1]
, the simplicial volume of M depends only on the image of its fundamental class under the classifying map.

The following inequality of M. Gromov [24, p. 37] connects the minimal volume entropy of an orientable connected closed manifold to its simplicial volume (see [START_REF] Sabourau | Small volume of balls, large volume entropy and the Margulis constant[END_REF] for other topological conditions ensuring the positivity of the minimal volume entropy through a different approach). Namely, every orientable connected closed m-manifold M satisfies

Ω(M ) ≥ c m M ∆ (1.6)
for some positive constant c m depending only on m. Extending this inequality to the semi-norm level, see Theorem 4.12, we obtain that every homology class a ∈ H m (X; R) of a path-connected topological space X satisfies a E ≥ c m a ∆ with the same constant c m as in (1.6).

A central question regarding the metrization of homotopy theory is to compare two given semi-norms in homology; see [27, §5.41]. In particular, one can ask whether a reverse inequality to (1.6) holds.

The following result affirmatively answers this question.

Theorem 1.3. Let m be a positive integer. Then there exist two positive constants c m and C m such that every homology class a ∈ H m (X; R) of a path-connected topological space X satisfies

c m a ∆ ≤ a E ≤ C m a ∆ .
We immediately deduce the following corollary.

Corollary 1.4. Let X be a path-connected topological space and a ∈ H m (X; R) be a homology class. Then a E vanishes if and only if a ∆ vanishes.

In particular, for every orientable connected closed manifold M , the volume entropy seminorm M E is zero if and only if the simplicial semi-norm M ∆ is zero.

In relation with Corollary 1.4, note that we do not know whether the volume entropy of an orientable connected closed manifold with zero simplicial volume necessarily vanishes. See [START_REF] Babenko | Minimal volume entropy and fiber growth[END_REF] for polyhedral counterexamples.

In this article, we also introduce the systolic volume semi-norm, whose definition rests on a new asymptotically optimal estimate in systolic geometry; see Theorem 1.5. Before stating this result, we need to introduce various notions.

Let M be a closed m-dimensional manifold or pseudomanifold with a (piecewise) Riemannian metric g. Let f : M → X be a map to a topological space X. The systole of M relative to f , denoted by sys f (M, g), is defined as the least length of a loop γ in M whose image by f is noncontractible in X. The systolic volume of M relative to f is defined as

σ f (M ) = inf g vol(M, g) sys f (M, g) m (1.7)
where the infimum is taken over all (piecewise) Riemannian metrics g on M . When f : M → X is π 1 -injective, for instance, when f : M → K(π 1 (M ), 1) is the classifying map of M , we simply denote its systolic volume by σ(M ) without any reference to f . By [START_REF] Babenko | Topologie des systoles unidimensionnelles[END_REF], [START_REF] Babenko | Addenda à l'article intitulé "Topologie des systoles unidimensionnelles[END_REF], [START_REF] Brunnbauer | Homological invariance for asymptotic invariants and systolic inequalities[END_REF], the systolic volume of a closed orientable manifold depends only on the image of its fundamental class under the classifying map.

As with (1.2), the systolic volume of a homology class a ∈ H m (X; Z), where X is a pathconnected topological space, is defined as

σ(a) = inf (M,f ) σ f (M ) (1.8)
where the infimum is taken over all m-dimensional geometric cycles (M, f ) representing a.

Let us present some known estimates on the systolic volume. There exist two positive constants A and B such that every closed genus g surface Σ g satisfies

A g (log g) 2 ≤ σ(Σ g ) ≤ B g (log g) 2 .
The first inequality was established by M. Gromov [START_REF] Gromov | Filling Riemannian manifolds[END_REF], [START_REF] Gromov | Systoles and intersystolic inequalities[END_REF]. The second inequality was proved by P. Buser and P. Sarnak in [START_REF] Buser | On the period matrix of a Riemann surface of large genus. With an appendix[END_REF], where they constructed hyperbolic genus g surfaces with a systole roughly equal to log(g).

In higher dimension, M. Gromov [START_REF] Gromov | Filling Riemannian manifolds[END_REF], [START_REF] Gromov | Systoles and intersystolic inequalities[END_REF] related the systolic volume σ(a) of a homology class a ∈ H m (X; Z) to its simplicial volume a ∆ through the following lower bound

σ(a) ≥ λ m a ∆ (log(2 + a ∆ )) m (1.9)
where λ m is a positive constant depending only on m. In particular, for every a ∈ H m (X; Z) with nonzero simplicial volume, we have

σ(k a) ≥ λ k (log k) m (1.10)
where λ = λ(a) > 0.

In a different direction, one can ask for an asymptotic upper bound on σ(k a). This problem was considered in [START_REF] Babenko | Géométrie systolique des sommes connexes et des revêtements cycliques[END_REF], where a sublinear upper bound in k was established, and in [START_REF] Babenko | Systolic volume of homology classes[END_REF], where the upper bound was improved.

Using different techniques, we obtain an asymptotically optimal upper bound on σ(k a). When the simplicial volume of a is nonzero, this upper bound shows that the lower bound (1.10) is roughly optimal in k, which positively answers a conjecture of [START_REF] Babenko | Systolic volume of homology classes[END_REF].

Theorem 1.5. Let X be a path-connected topological space. Then for every homology class a ∈ H m (X; Z), there exists a constant C = C(a) > 0 such that for every integer k ≥ 2, we have the distance from the origin and apply a stabilization process to σ as in (1.5). Namely, for every a ∈ H m (X; Z) with m ≥ 3, define .11) This functional extends to H m (X; R) H m (X; Z) ⊗ R in a canonical way and gives rise to a semi-norm, still denoted by • σ , on H m (X; R), called the systolic volume semi-norm. Note that this definition differs from the one proposed in [27, §5.41].

σ(k a) ≤ C k (log k) m .
a σ = lim k→∞ σ(k a) k . ( 1
For an orientable connected closed m-manifold M , define

M σ = [M ] σ where [M ] ∈ H m (M ; Z) is the fundamental class of M .
The systolic volume semi-norm satisfies similar functorial properties to the volume entropy semi-norm and the simplicial volume semi-norm; see Theorem 6.4.

It follows from (1.9) that every homology class a ∈ H m (X; Z) of a path-connected topological space X satisfies a σ ≥ λ m a ∆ with the same positive constant λ m as in (1.9); see Section 7 for further detail and an alternate approach based on a comparison between the systolic volume semi-norm and the volume entropy semi-norm.

As previously, we show that the systolic volume semi-norm and the simplicial volume seminorm are equivalent in homology.

Theorem 1.6. Let m ≥ 3 be an integer. Then there exist two positive constants λ m and µ m such that every homology class a ∈ H m (X; R) of a path-connected topological space X satisfies

λ m a ∆ ≤ a σ ≤ µ m a ∆ .
Theorem 1.6 contrasts with the existence of a sequence of closed m-manifolds (e.g., closed hyperbolic 3-manifolds) with bounded simplicial volume and arbitrarily large systolic volume; see [START_REF] Sabourau | Systolic volume of hyperbolic manifolds and connected sums of manifolds[END_REF]. This illustrates the effect the double stablization process can have on the systolic volume by significantly lowering its value.

Combining the recent result [START_REF] Heuer | The spectrum of simplicial volume[END_REF] on the spectrum of the simplicial volume with Theorem 1.3, we immediately deduce that the volume entropy semi-norm and the systolic volume semi-norm are not bounded away from zero in dimension greater than 3. (In dimension 2 and 3, there is a gap in the simplicial volume spectrum, and so in the volume entropy spectrum and the systolic volume spectrum by Theorem 1.3 and Theorem 1.6.) More generally, we have the following result.

Corollary 1.7. Let m ≥ 4 be an integer. Then the sets of all volume entropy semi-norms M E and of all systolic volume semi-norms M σ , where M is an orientable connected closed mmanifold, are dense in [0, ∞).

In his book [27, §5.41], M. Gromov suggests to study some functionals of geometric nature in homology. These functionals measure the minimal volume of a singular Riemannian manifold representing a given homology class with some constraint on the metric. After a stabilization process as in (1.5) or (1.11), they should give rise to homology semi-norms. Our definitions of the volume entropy semin-norms and the systolic volume semi-norm are inspired by this general idea. However, they differ from the constructions sketched in [27, p. 310-311], which do not consider relative volume entropy or relative systole and lead to a number of technical difficulties.

Articles about minimal volume entropy closely related to our paper include [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF], [START_REF] Babenko | Topologie des systoles unidimensionnelles[END_REF], [START_REF] Babenko | Addenda à l'article intitulé "Topologie des systoles unidimensionnelles[END_REF], [START_REF] Besson | Volume et entropie minimale des espaces localement symétriques[END_REF], [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF], [START_REF] Brunnbauer | Homological invariance for asymptotic invariants and systolic inequalities[END_REF], [START_REF] Gromov | Volume and bounded cohomology[END_REF], [START_REF] Merlin | Minimal entropy for uniform lattices in product of hyperbolic planes[END_REF], [START_REF] Sabourau | Systolic volume and minimal entropy of aspherical manifolds[END_REF], [START_REF] Sabourau | Small volume of balls, large volume entropy and the Margulis constant[END_REF], [START_REF] Sambusetti | Minimal entropy and simplicial volume[END_REF], [START_REF] Sambusetti | On minimal entropy and stability[END_REF]. Connections between the systolic volume and the minimal volume entropy can be found in [START_REF] Katz | Entropy of systolically extremal surfaces and asymptotic bounds[END_REF], [START_REF] Sabourau | Systolic volume and minimal entropy of aspherical manifolds[END_REF] and [START_REF] Brunnbauer | Homological invariance for asymptotic invariants and systolic inequalities[END_REF].

This article is organized as follows. In Section 2, we establish lower and upper bounds on the minimal volume entropy of the connected sum of closed manifolds, and derive that the functional • E is a semi-norm in real homology. Functorial properties of the volume entropy semi-norm are presented in Section 3. In Section 4, we show that the volume entropy semi-norm of a homology class is bounded from above and below by its simplicial volume, up to some multiplicative constants depending only on the degree of the homology class. Therefore, the volume entropy semi-norm and the simplicial volume are equivalent homology semi-norms. Our approach for the upper bound relies on a geometrization of the simplicial volume and the universal realization of homology classes established by A. Gaifullin [START_REF] Gaifullin | Realization of cycles by aspherical manifolds[END_REF], [START_REF] Gaifullin | The manifold of isospectral symmetric tridiagonal matrices and the realization of cycles by aspherical manifolds[END_REF] regarding Steenrod's problem. More than the result about the universal realization of homology classes, we will need to retrieve combinatorial features of the construction to apply our argument leading to an upper bound on the volume entropy semi-norm of a homology class. The reverse inequality is obtained through the use of bounded cohomology by adapting M. Gromov's chain diffusion technique. In Section 5, we bound from above the systolic volume of the multiple of a given homology class.

The proof relies on topological properties of the universal realizators in homology used in the previous section and on systolic estimates in geometric group theory. This optimal asymptotic estimate allows us to define the systolic volume semi-norm in Section 6. Functorial properties and comparison results of the systolic volume (semi-norm) are also presented. In Section 7, we show that the systolic volume semi-norm and the simplicial volume semi-norm are equivalent in every homology degree. In the last section, we derive the density of the volume entropy and systolic volume semi-norm spectra in dimension at least four.

Acknowledgment. The second author thanks the Fields Institute and the Department of Mathematics at the University of Toronto for their hospitality while part of this work was carried out. The authors thank the referees for their relevant comments, which helped improve the exposition.

Entropy of connected sums

In this section, we first establish an additive formula of the functional Ω for the bouquet of simplicial complexes. We also obtain lower and upper bounds on the minimal volume entropy of the connected sum of two closed manifolds, and derive that the functional • E is a semi-norm in real homology. Finally, we present a couple of applications of these estimates.

Preliminaries.

Let us first recall the definition of a pseudomanifold. Definition 2.1. A connected closed m-dimensional pseudomanifold is a finite simplicial complex M such that (1) every simplex of M is a face of some m-simplex of M ;

(2) every (m -1)-simplex of M is the face of exactly two m-simplices of M ;

(3) given two m-simplices s and s of M , there exists a finite sequence s = s 1 , s 2 , . . . , s n = s of m-simplices of M such that s i and s i+1 have an (m -1)-face in common. The m th homology group H m (M ; Z) of a connected closed m-dimensional pseudomanifold is either isomorphic to Z or trivial; see [START_REF] Spanier | Algebraic topology[END_REF]. In the former case, we say that the pseudomanifold M is orientable.

Consider a finite simplicial complex K with a piecewise Riemannian metric g (also called polyhedral Riemannian metric). Denote by ρ the distance induced by g on K and on all the coverings of K. Let H ¡G where G = π 1 (K). The quotient group G/H acts by isometries on the H-covering K H . Furthermore, the action of G/H on K H is proper, discontinuous, without any fixed point. Fix q ∈ K H . The orbit of q under the action of G/H on K H is denoted by q •(G/H). Let also B H (t, q; g) = {x ∈ K H | ρ(q, x) ≤ t} be the ball of radius t centered at q in K H .

The volume entropy of K relative to H is equal to the exponential growth rate of the number of points in the orbit of q under G/H, as stated in the following classical result; see [START_REF] Sabourau | Systolic volume and minimal entropy of aspherical manifolds[END_REF]Lemma 2.3] for instance. Proposition 2.2. Let K be a finite simplicial complex with a piecewise Riemannian metric. Let H ¡ G where G = π 1 (K). Then Let us recall a few results established in [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF]; see also [START_REF] Sabourau | Systolic volume and minimal entropy of aspherical manifolds[END_REF]Lemma 3.5].

ent H (K, g) = lim t→∞ 1 t log |B H (t, q; g) ∩ q • (G/H)|. ( 2
Definition 2.3. A simplicial map f : K 1 → K 2 between two m-dimensional simplicial complexes is m-monotone if for every point x 2 in the interior of an m-simplex of K 2 , the preimage f -1 (x 2 ) is connected (and so is either empty or a singleton).

We will need the following comparison principle proved in [1, §2].

Proposition 2.4. For i = 1, 2, let K i be an m-dimensional simplicial complex and φ i : π 1 (K i ) → G be an epimorphism. Suppose that there exists an m-monotone map f :

K 1 → K 2 such that φ 1 = φ 2 • f * . Then Ω H 1 (K 1 ) ≤ Ω H 2 (K 2 ) where H i = ker φ i .
Actually, Proposition 2.4 is a straighforward consequence of the following result proved in [1, §2] and [2, Lemme 3.1], which will also be used in the sequel. Lemma 2.5. Let f : K 1 → K 2 be an m-monotone map between two m-dimensional simplicial complexes. Then for every polyhedral Riemannian metric g on K 2 and every ε > 0, there exists a polyhedral Riemannian metric g ε on K 1 with

vol(K 1 , g ε ) ≤ vol(K 2 , g) + ε such that f is nonexpanding.
Let G be a finitely presented group. For every subgroup H of G, denote by H the normal closure of H in G.

The following result provides a formula for the minimal volume entropy of the bouquet of two simplicial complexes.

Theorem 2.6. Let m ≥ 2. For i = 1, 2, let K i be a connected m-dimensional simplicial complex and H i ¡ π 1 (K i ) be a normal subgroup. Then

Ω H 1 * H 2 (K 1 ∨ K 2 ) = Ω H 1 (K 1 ) + Ω H 2 (K 2 ) (2.2)
where the basepoint of the bouquet K 1 ∨ K 2 is a vertex.

Proof. First we prove the inequality

Ω H 1 (K 1 ) + Ω H 2 (K 2 ) ≤ Ω H 1 * H 2 (K 1 ∨ K 2 ). (2.3) Let K = K 1 ∨ K 2 .
By van Kampen's theorem [29, §1.2], we have

π 1 (K) π 1 (K 1 ) * π 1 (K 2 ).
Let g be a polyhedral Riemannian metric on K and g i be its restriction to K i for i = 1, 2. Let K i and K be the normal covers corresponding to the normal subgroups

H i ¡ π 1 (K i ) and H 1 * H 2 ¡ π 1 (K)
, with the lifted metrics g i and g. Observe that the canonical inclusions

K i ⊆ K are isometric. This implies ent H i (K i , g i ) ≤ ent H 1 * H 2 (K, g).
Thus, for every i = 1, 2,

Ω H i (K i ) ≤ ent H i (K i , g i ) m vol(K i , g i ) ≤ ent H 1 * H 2 (K, g) m vol(K i , g i ).
Adding the two inequalities so-obtained for i = 1, 2, and using the relation

vol(K, g) = vol(K 1 , g 1 ) + vol(K 2 , g 2 )
we finally derive

Ω H 1 (K 1 ) + Ω H 2 (K 2 )
≤ Ω H 1 * H 2 (K, g) for every polyhedral Riemannian metric g on K. Hence the inequality (2.3). Now, let us prove the reverse inequality

Ω H 1 * H 2 (K 1 ∨ K 2 ) ≤ Ω H 1 (K 1 ) + Ω H 2 (K 2 ).
(2.4) We proceed in two steps. Without loss of generality, we can assume that the two subcomplexes K 1 and K 2 of K are glued at a common vertex. Let

p i ∈ K i be a vertex such that K 1 ∨ K 2 = K 1 ∪ p 1 =p 2 K 2 .
Define the m-dimensional simplicial complex

P = K 1 ∪ p 1 ={1} [1, 2] ∪ {2}=p 2 K 2 . (2.5)
For the first step, let us show that

Ω H 1 * H 2 (P ) = Ω H 1 * H 2 (K 1 ∨ K 2 ).
Contracting the interval [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF][START_REF] Babenko | Topologie des systoles unidimensionnelles[END_REF] in P to a point gives rise to an m-monotone simplicial map

P -→ K 1 ∨ K 2 inducing a π 1 -isomorphism. By Proposition 2.4, we derive Ω H 1 * H 2 (P ) ≤ Ω H 1 * H 2 (K 1 ∨ K 2 ). (2.6)
For the reverse inequality, let θ i be a triangulation of K i for every i = 1, 2. Denote by St(p i ) the open star of p i for the triangulation θ i . Let θ i be the triangulation of K i which agrees with θ i in K i \ St(p i ) and with the semi-barycentric triangulation of St(p i ) in St(p i ) (obtained by adding a vertex at the barycenter of every simplex of St(p i )). The bouquet K 1 ∨ K 2 is endowed with the triangulations given by θ 1 and θ 2 . The complex P = K 1 ∪ [1, 2] ∪ K 2 is endowed with the triangulation given by θ 1 , θ 2 and the barycentric subdivision of [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF][START_REF] Babenko | Topologie des systoles unidimensionnelles[END_REF] into

I 1 = [1, 3 2 ] and I 2 = [ 3 2 , 2]
. Consider the simplicial map f : K 1 ∨ K 2 → P which agrees with the identity map on K i \St(p i ), and takes p i to the midpoint of [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF][START_REF] Babenko | Topologie des systoles unidimensionnelles[END_REF] and all the vertices of θ i corresponding to the barycenters of the simplices of St(p i ) for the triangulation θ i to i. By construction, the map f is m-monotone and induces a π 1 -isomorphism.

The inequality obtained by applying Proposition 2.4 to f , combined with the inequality (2.6), yields the relation

Ω H 1 * H 2 (P ) = Ω H 1 * H 2 (K 1 ∨ K 2 ).
(2.7) For the second step, we need to show that

Ω H 1 * H 2 (P ) ≤ Ω H 1 (K 1 ) + Ω H 2 (K 2 ).
(2.8)

Fix β i > Ω H i (K i ) 1 m
. By definition, there exists a metric

h i on K i such that ent H i (K i , h i ) m vol(K i , h i ) < β m i .
(2.9)

By scale invariance, this inequality holds for every homothetic metric λ 2 i h i with λ i > 0. Choose the factors λ 1 and λ 2 so that ent

H 1 (K 1 , λ 2 1 h 1 ) = ent H 2 (K 2 , λ 2 2 h 2 ) (2.10) and vol(K 1 , λ 2 1 h 1 ) + vol(K 2 , λ 2 2 h 2 ) = 1. (2.11) Let α = ent H 1 (K 1 , λ 2 1 h 1 ) = ent H 2 (K 2 , λ 2 2 h 2 )
. The relations (2.9) and (2.10) combined with (2.11) show that

α m < β m 1 + β m 2 .
(2.12) Consider the metric g d on P which is defined on the three parts of P given by (2.5) as follows

g d =      λ 2 1 h 1 on K 1 4d 2 dx 2 on [p 1 , p 2 ] λ 2 2 h 2 on K 2 (2.13)
where x is the coordinate on [p 1 , p 2 ] = [1, 2] and d > 0 is a parameter. By construction, we have length g d ([p 1 , p 2 ]) = 2d and vol(P, g d ) = 1, where the second equality comes from (2.11).

We will need the following result.

Lemma 2.7. Let ε > 0. For d large enough, we have

ent H 1 * H 2 (P, g d ) < α + ε.
Proof. Let K i and P be the normal covers corresponding to the normal subgroups

H i ¡ π 1 (K i ) and H 1 * H 2 ¡ π 1 (P ). The cover P of P = K 1 ∪ [p 1 , p 2 ]
∪ K 2 can be described as follows

(1) the cover P decomposes into the union of the lifts of the subsets K 1 and K 2 of P , also called leaves of P , and the lifts of [p 1 , p 2 ]; (2) every lift of [p 1 , p 2 ] in P is adjacent to two leaves homeomorphic to K 1 and K 2 ;

(3) removing a lift of [p 1 , p 2 ] from P separates the cover into two connected components. The group G = π 1 (P )/ H 1 * H 2 where

π 1 (P ) π 1 (K 1 ∨ K 2 ) π 1 (K 1 ) * π 1 (K 2 ) decomposes into G G 1 * G 2 , where G i = π 1 (K i )/H i (
this relation is left to the reader as an exercice in group theory). With this decomposition, the action of G on P can be described as follows. Let F K i be a leaf of P . The subgroup G i = π 1 (K i )/H i of G acts on F ⊆ P . For every lift q i of p i in F , the orbit q i • G i of q i in F is composed of all the lifts of p i lying in F under the cover P → P .

Denote by ρ i the distance on K i induced by λ 2 i h i and denote by g d the metric on P induced by g d ; see (2.13). Let [q 1 , q 2 ] be a lift of [p 1 , p 2 ] in P and q be the midpoint of [q 1 , q 2 ]. In view of (2.1), the desired bound on ent

H 1 * H 2 (P, g d ) will follow from a bound on v(t; d) = |V (t; d)|, where V (t; d) = B H 1 * H 2 (t, q; g d ) ∩ (q • G). (2.14)
By the normal form theorem for free product of groups, see [START_REF] Kurosh | The theory of Groups[END_REF], every element γ ∈ G G 1 * G 2 can be uniquely written in normal form as

γ = γ 1 γ 2 . . . γ l (2.15)
where γ s is a nontrivial element of G 1 or G 2 for s ∈ {1, . . . , l}, and γ s and γ s+1 do not lie in the same factor G 1 or G 2 for s ∈ {1, . . . , l -1}. For s ∈ {1, . . . , l}, denote by i s ∈ {1, 2} the index such that γ s ∈ G is . The length l(γ) of γ is the number l of elements in the decomposition (2.15). It follows from the description of the cover P , see ( 1)-( 3), and of the action of G i on every leaf F K i of P that every path from q to q • γ, where γ is of length l, passes through the points

q, . . . , q is • (γ s-1 . . . γ 1 ), q is • (γ s . . . γ 1 ), q s+1 • (γ s+1 . . . γ 1 ), . . . , q • γ
where s runs over {1, . . . , l -1}. Indeed, removing any of these points between q and q • γ from P disconnects q and q •γ. Since G acts by isometries on P , the g d -distance between q is •(γ 1 . . . γ s-1 ) and

q is • (γ 1 . . . γ s ) is equal to dist g d (q is , q is • γ s ).
Since the restriction of the distance dist g d to a leaf F K i of P agrees with the distance ρ i on K i and since the action of G i on F as a subgroup of G coincides with its action on K i under the identification F K i , we have

dist g d (q is , q is • γ s ) = ρ i (p is , p is • γ s ). Thus, dist g d (q, q • γ) = d + ρ i 1 (p i 1 , p i 1 • γ 1 ) + 2d + ρ i 2 (p i 2 , p i 2 • γ 2 ) + 2d + • • • + ρ i l (p i l , p i l • γ l ) + d. Hence, dist g d (q, q • γ) = 2dl + l s=1 ρ is (p is , p is • γ s ).
(2.16)

To estimate the exponential growth rate of the orbit of G in P , it will be useful to decompose G by the filtration induced by the length on G. Under this filtration, the group G decomposes into the disjoint union

G = ∞ l=1 G (l)
where G (l) is formed of the elements of G of length l. We derive from (2.14) that

V (t; d) = l≥1 V (l) (t; d) (2.17)
where

V (l) (t; d) = B H 1 * H 2 (t, q; g d ) ∩ (q • G (l) ).
Since the union (2.17) is disjoint, we can write

v(t; d) = l≥1 v (l) (t; d) where v (l) (t; d) = |V (l) (t; d)|. Suppose q • γ ∈ V (l) (t; d). Let t s be the smallest integer greater or equal to ρ is (p is , p is • γ s ). Then p is • γ s ∈ V is (t s ), where V i (t) = B H i (t, p i ; ρ i ) ∩ (p i • G i ) . Furthermore, t s < ρ is (p is , p is • γ s ) + 1. By (2.16), this inequality leads to l s=1 t s < l s=1 ρ is (p is , p is • γ s ) + l < t -2dl + l = t -(2d -1)l. (2.18) Therefore, every element γ ∈ G with q • γ ∈ V (t; d) decomposes into a product γ = γ 1 . . . γ l with γ s ∈ G is , see (2.15), such that p is • γ s ∈ V is (t s )
, where the integers t s defined from γ s satisfy (2.18). The number of elements γ ∈ G of length l with q • γ ∈ V (t; d) and given integers t s satisfying (2.18) is at most

|V i 1 (t 1 )| • |V i 2 (t 2 )| • • • |V i l (t l )|.
By definition of α, see below (2.11), the exponential growth rate of |V is (t)| agrees with α. Thus, for every α 1 > α (arbitrarily close to α, which will be specified afterwards), there exists t 0 > 0 such that |V is (t)| < e α 1 t for every t > t 0 .

Let I be the subset of L = {1, . . . , l} given by

I = {s ∈ L | t s ≤ t 0 }. Let C = max{V 1 (t 0 ), V 2 (t 0 )}. For every s ∈ L, we have |V is (t s )| ≤ C if s ∈ I |V is (t s )| ≤ e α 1 ts if s / ∈ I.
These estimates yield an upper bound on the product

|V i 1 (t 1 )| • |V i 2 (t 2 )| • • • |V i l (t l )| ≤ C |I| e α 1 s / ∈I ts ≤ C l e α 1 (t-(2d-1)l) (2.19)
where the last inequality follows from |I| ≤ l and the bound (2.18). Now, the number of l-uplets τ = (t 1 , t 2 , . . . , t l ) with nonnegative integral coordinates satisfying (2.18) is bounded by

[t -(2d -1)l] l l! .
Combined with (2.19), this leads to

v (l) (t; d) ≤ τ |V i 1 (t 1 )| • |V i 2 (t 2 )| • • • |V i l (t l )| ≤ C l e α 1 (t-(2d-1)l) [t -(2d -1)l] l l!
where τ runs over all l-uplets satisfying (2.18). For d > 1 2 , we have t -(2d -1)l ≤ t and so

v (l) (t; d) ≤ C l e α 1 t e -α 1 (2d-1)l t l l! = e α 1 t l! Cd e α 1 (2d-1) l t d l .
For d large enough, we have Cd e α 1 (2d-1) < 1.

(2.20) Thus,

v (l) (t; d) ≤ e α 1 t l! t d l .
Therefore,

v(t; d) = l≥1 v (l) (t; d) ≤ l≥1 e α 1 t l! t d l = e α 1 t+ t d .
Hence, Let us resume the proof of Theorem 2.6. Since β m i can be arbitrarily close to Ω H i (K i ), the inequality (2.12) combined with Lemma 2.7 leads to

lim t→∞ log v(t; d) t ≤ α 1 + 1 d . ( 2
Ω H 1 * H 2 (P ) ≤ Ω H 1 (K 1 ) + Ω H 2 (K 2 ).
Along with (2.7), this inequality yields the desired result.

Upper bound on the minimal volume entropy of connected sums.

The following result compares the minimal volume entropy of the connected sum of two (pseudo)manifolds with the minimal volume entropy of the two (pseudo)manifolds. Here, the connected sum of the two connected m-pseudomanifolds is defined in the usual way by removing an m-simplex ∆ m from each pseudomanifold and by identifying the boundary ∂∆ m of the resulting pseudomanifolds.

Theorem 2.8. For i = 1, 2, let M i be a connected closed pseudomanifold of dimension m ≥ 3 and

H i ¡ π 1 (M i ) be a normal subgroup. Then Ω H 1 * H 2 (M 1 M 2 ) ≤ Ω H 1 (M 1 ) + Ω H 2 (M 2 ).
(2.22)

Remark 2.9. In dimension 2, the result remains valid by replacing the normal subgroup

H 1 * H 2 with f -1 ( H 1 * H 2 ), where f : M 1 M 2 -→ M 1 ∨ M 2 is the canonical projec- tion.
Remark 2.10. Inequality (2.22) is the analogue for the volume entropy of a similar bound holding for the systolic volume; see [5, Proposition 3.6] and Proposition 6.6. Note however that the proof is more intricate with the minimal volume entropy than with the systolic volume.

Proof of Theorem 2.8. Consider the canonical m-monotone map

f : M 1 M 2 → M 1 ∨ M 2
obtained by collapsing the attaching sphere to a point (in order to get a simplicial map, we may have to take two barycentric subdivisions of M 1 and M 2 ). Since m ≥ 3, the induced homomorphism f * :

π 1 (M 1 M 2 ) → π 1 (M 1 ∨ M 2
) is an isomorphism. The comparison principle, see Proposition 2.4, and Theorem 2.6 yield

Ω H 1 * H 2 (M 1 M 2 ) ≤ Ω H 1 * H 2 (M 1 ∨ M 2 ) = Ω H 1 (M 1 ) + Ω H 2 (M 2 ).
Corollary 2.11. Let X be a path-connected topological space. Then for every a 1 , a 2 ∈ H m (X; Z), we have Ω(a 1 + a 2 ) ≤ Ω(a 1 ) + Ω(a 2 ). In particular, the quantity Ω(a 1 -a 2 ) defines a pseudo-distance between a 1 and a 2 in H m (X; Z), and the functional Furthermore, by Theorem 2.8, we have

• E is a semi-norm on H m (X; R). Proof. Let a 1 , a 2 ∈ H m (X; R). Fix ε > 0. There exists a map f i : M i → X from an oriented connected closed m-pseudomanifold M i representing a i for i = 1, 2 such that Ω ker(f i ) * (M i ) ≤ Ω(a i ) + ε. Let M = M 1 M 2 . Consider the canonical map f = f 1 ∨ f 2 : M → X obtained from f
Ω kerf * (M ) ≤ Ω ker(f 1 ) * (M 1 ) + Ω ker(f 2 ) * (M 2 )
≤ Ω(a 1 ) + Ω(a 2 ) + 2ε

Hence, Ω(a 1 + a 2 ) ≤ Ω(a 1 ) + Ω(a 2 ). Replacing a i with k a i in the previous inequality, dividing by k and letting k go to infinity, we obtain

a 1 + a 2 E ≤ a 1 E + a 2 E .
Since • E is clearly homogeneous by the stabilization process, see (1.5), the functional • E is a semi-norm.

2.4. Lower bound on the minimal volume entropy of connected sums.

In a different direction, taking the connected sum with an orientable pseudomanifold does not decrease the minimal volume entropy as the following result shows. Theorem 2.12. For i = 1, 2, let M i be a connected closed pseudomanifold of dimension m ≥ 3 and

H i ¡ π 1 (M i ) be a normal subgroup. Let H ¡ π 1 (M 1 ) * π 1 (M 2 ) be a normal subgroup such that the canonical inclusion π 1 (M 1 ) π 1 (M 1 ) * π 1 (M 2 ) induces an inclusion π 1 (M 1 )/H 1 (π 1 (M 1 ) * π 1 (M 2 ))/H.
(2.23)

Suppose M 2 is orientable.
Then

Ω H 1 (M 1 ) ≤ Ω H (M 1 M 2 ). (2.24)
Combined with Theorem 2.8, we obtain Corollary 2.13. For i = 1, 2, let M i be a connected closed pseudomanifold of dimension m ≥ 3 and

H i ¡ π 1 (M i ) be a normal subgroup. Suppose M 2 is orientable and Ω H 2 (M 2 ) = 0.
Then

Ω H 1 * H 2 (M 1 M 2 ) = Ω H 1 (M 1 ).
In order to prove Theorem 2.12, we first establish the following result. For a CW -complex X, denote by X(k) its k-skeleton. Proposition 2.14. Let M be an orientable connected closed pseudomanifold of dimension m ≥ 3.

Suppose that

M = D m ∪ φ M (m -1) (2.25)
is a cell decomposition with a single m-cell. Then the space

M ∪ M (m-2) Cone(M (m -2))
obtained by gluing the cone

Cone(M (m -2)) over M (m -2) to M along M (m -2) is homotopy equivalent to a finite bouquet of spheres M ∪ M (m-2) Cone(M (m -2)) s S m-1 s S m .
Proof. We have

M ∪ M (m-2) Cone(M (m -2)) M/M (m -2) s S m-1 s ∪ φ S m
where the number of (m -1)-spheres S m-1 s is equal to the number of (m -1)-cells of M and

φ : S m-1 φ -→ M (m -1) -→ M (m -1)/M (m -2) s S m-1 s
is the projection of the attaching map φ.

To derive the proposition, we need to show that φ is null-homotopic. Consider the triple (M, M (m -1), M (m -2)) and the corresponding long exact sequence with Z-coefficients

. . . 0 i * -→ H m (M, M (m -2)) j * -→ H m (M, M (m -1)) ∂ -→ H m-1 (M (m -1), M (m -2)) -→ . . . .
From the long exact sequence of the pair (M, M (m -2)), we obtain that

H m (M, M (m -2)) H m (M ).
The orientability of M implies that

H m (M, M (m -1)) H m (M ).
Thus, j * is an isomorphism, which implies that ∂ = 0.

Thinking of the homology groups We can now prove Theorem 2.12.

Proof of Theorem 2.12. Choose a cell decomposition of M 2 with only one cell of maximal dimension m, which is coherent with the triangulation of the pseudomanifold. Denote by Cone(M 2 (m -2)) the cone over the (m -2)-skeleton M 2 (m -2) of M 2 . By Proposition 2.14, the space

M 2 ∪ M 2 (m-2) Cone(M 2 (m -2)) obtained by gluing the cone Cone(M 2 (m -2)) to M 2 along M 2 (m -2) is homotopy equivalent to a finite bouquet of spheres M 2 ∪ M 2 (m-2) Cone(M 2 (m -2)) s S m-1 s S m
with only one m-dimensional sphere S m . Thus, the canonical inclusion M 1 \ B m ⊆ M 1 M 2 extends to the missing ball B m and gives rise to an m-monotone map, see Definition 2.3,

f : M 1 → M 1 M 2 ∪ M 2 (m-2)
Cone(M 2 (m -2)).

(2.26)

Fix ε > 0. Consider a metric g on M 1 M 2 with vol(M 1 M 2 , g) = 1, which is ε-extremal, that is, ent H (M 1 M 2 , g) m ≤ Ω H (M 1 M 2 ) + ε. (2.27)
Extend the metric g to a metric g on

M 1 M 2 ∪ M 2 (m-2) Cone(M 2 (m-2)) as follows. First, observe that Cone(M 2 (m -2)) = M 2 (m -2) × [0, 1]/M 2 (m -2) × {1}. The extension g of g, which agrees with g on M 1 M 2 , is defined on Cone(M 2 (m -2)) by g = g |M 2 (m-2) + 10D dt 2 if 0 ≤ t ≤ 1 2 4(1 -t) 2 g |M 2 (m-2) + 10D dt 2 if 1 2 ≤ t ≤ 1 where D = diam M 2 (m -2), g |M 2 (m-2)
. Technically, the metric g is singular, but it still induces a distance dist g . Note also that by dimensional reasons

vol(M 1 M 2 ∪ M 2 (m-2) Cone(M 2 (m -2)), g ) = vol(M 1 M 2 , g) = 1.
By construction of g , the canonical inclusion

i : M 1 M 2 → M 1 M 2 ∪ M 2 (m-2) Cone(M 2 (m -2)) (2.28) is distance-preserving. That is, for every p 1 , p 2 ∈ M 1 M 2 , we have dist g (p 1 , p 2 ) = dist g (i(p 1 ), i(p 2 )).
Since m ≥ 3, the composite map

M 1 \ B m ⊆ M 1 M 2 → M 1 M 2 ∪ M 2 (m-2) Cone(M 2 (m -2)) (2.29)
induces an isomorphism between the fundamental groups. Denote by

G 1 G = π 1 (M 1 M 2 ) the image of π 1 (M 1 ) in π 1 (M 1 M 2 ).
Let

q 1 ∈ M 1 \ B m ⊆ M 1 M 2 . Since Cone(M 2 (m -2)
) is simply connected and the map (2.29) is distance-preserving, every loop γ

⊆ M 1 M 2 ∪ M 2 (m-2)
Cone(M 2 (m-2)) based at q 1 is homotopic to a loop γ ⊆ M 1 M 2 based at the same point such that length g (γ ) ≤ length g (γ).

(2.30)

The group G/H acts on the cover M of M 1 M 2 with fundamental group H. Similarly, the group

G 1 /H 1 acts on the cover M of M 1 M 2 ∪ M 2 (m-2)
Cone(M 2 (m -2)) with fundamental group H 1 . Let g and g be the metrics on M and M induced by g and g . Fix some lifts q ∈ M and q ∈ M of q 1 . Denote by B H (t, q; g) and B H 1 (t, q ; g ) the balls of M and M of radius t centered at q and q . Since G 1 /H 1 G/H, see (2.23), it follows from (2.30) that for every t ≥ 0

|B H 1 (t, q ; g ) ∩ q • (G 1 /H 1 )| ≤ |B H (t, q; g) ∩ q • (G/H)|. (2.31)
Applying Lemma 2.5 to the m-monotone map f , see (2.26), we derive a polyhedral Riemannian metric g ε on M 1 such that the map f is nonexpanding and

vol(M 1 , g ε ) < vol(M 1 M 2 , g) + ε. (2.32)
The group G 1 /H 1 acts both on M and on the cover M of M 1 with fundamental group H 1 . Let g ε be the metric on M induced by g ε . Fix a lift q of q 1 in M . The nonexpanding map f lifts to a (G 1 /H 1 )-equivariant, nonexpanding map f : M → M . This implies that the ball B H 1 (t, q ; g ε ) of M satisfies

|B H 1 (t, q ; g ε ) ∩ q • (G 1 /H 1 )| ≤ |B H 1 (t, q ; g ) ∩ q • (G 1 /H 1 )|.
(2.33)

Combining the bounds (2.31) and (2.33), we derive the following inequalities on the exponential growth rates of the orbits of G 1 /H 1 and G/H

ent H 1 (M 1 , g ε ) ≤ ent H 1 (M 1 M 2 ∪ M 2 (m-2) Cone(M 2 (m -2)), g ) ≤ ent H (M 1 M 2 , g) Since vol(M 1 M 2 , g) = 1 and vol(M 1 , g ε ) < 1 + ε, see (2.32
), this estimate combined with (2.27) yields the desired bound

Ω H 1 (M 1 ) ≤ Ω H (M 1 M 2 ).
Remark 2.15. The proof of Theorem 2.12 does not apply when M 2 is nonoriented. The conclusion is unclear in this case.

Fundamental class of finite order.

The following result is a direct application of Theorem 2.12. Remark 2.17. In order to contextualize this result, recall that under the assumption of Proposition 2.

(i * ) m : H m (H; Q) → H m (G; Q).
Then for every homology class a ∈ H m (H; Z)

a E = i * (a) E .
Proof. Still denote by i : K(H, 1) → K(G, 1) the classifying map induced by the canonical inclusion i : H → G. By (1.3), for every integer k ≥ 1, we have

Ω(k i * (a)) = Ω(i * (k a)) ≤ Ω(k a).
Hence

a E ≥ i * (a) E .
Thus, we only have to show the converse inequality. The idea is to start with an almost extremal geometric cycle of K(G, 1) representing k i * (a), to add handles to it to make sure it is

M 2 = (S 1 × S m-1 ) (S 1 × S m-1 ) . . . (S 1 × S m-1 )
be the connected sum of p copies of S 1 × S m-1 . Let f 2 : M 2 → K(G, 1) be a map inducing a surjective homomorphism between the fundamental groups. Observe that Ω(M 2 ) = 0 and

(f 2 ) * ([M 2 ]) = 0 ∈ H m (G; Z).
Fix ε > 0. By definition of • E , for every integer k ≥ 1, there exists a map f 1 : M 1 → K(G, 1) defined on an oriented connected closed m-pseudomanifold representing the class k i * (a) ∈ H m (G; Z) such that

Ω ker(f 1 ) * (M 1 ) ≤ k ( i * (a) E + ε).
Consider the connected sum M = M 1 M 2 and the canonical map f = f 1 ∨ f 2 : M → K(G, 1) obtained from f 1 and f 2 by collapsing the attaching sphere into a point. Note that

ker f * ker(f 1 ) * * ker(f 2 ) * . Since (f 2 ) * ([M 2 ]) = 0 ∈ H m (G; Z), the map f : M → K(G, 1) still represents the class k i * (a) ∈ H m (G; Z).
Since Ω ker(f 2 ) * (M 2 ) = 0, we deduce from Corollary 2.13 that

Ω ker f * (M ) ≤ k ( i * (a) E + ε). (2.34)
Since f 2 induces a surjective homomorphism between the fundamental groups, the same holds for the map f . Let M be the cover of M of fundamental group f -1 * (H). Denote by

f : M → K(H, 1) the corresponding lift of f . Let b = f * ([ M ]) ∈ H m (H; Z). Since H is of index d in G, the cover π : M → M is of degree d. Thus, π * ([ M ]) = d [M ]. Still denote by i : K(H, 1) → K(G, 1) the classifying map induced by the canonical inclusion i : H → G. It follows from the commutation relation i • f = f • π that i * (b) = i * ( f * ([ M ])) = d f * ([M ]) = dk i * (a).
Since the canonical inclusion i : H → G induces a monomorphism between the m-dimensional rational homology groups, we deduce that b = dk a + c where c ∈ TorH m (H; Z). Thus,

b E = dk a E .
Let g be an ε-extremal metric on M , that is,

Ω ker f * (M, g) ≤ Ω ker f * (M ) + ε. (2.35)
Denote by g the lift of g on M . The cover of (M, g) of fundamental group ker f * is isometric to the cover of ( M , g) of fundamental group ker f * . Thus, the exponential growth rates of the volume of balls in the two coverings are equal. Since π : M → M is of degree d, we have vol( M ) = d vol(M ). Therefore,

Ω ker f * ( M , g) = d Ω ker f * (M, g).
Now, by construction, ( M , f ) represents b. Hence,

dk a E = b E ≤ Ω ker f * ( M , g) = d Ω ker f * (M, g).
This inequality combined with the bounds (2.34) and (2.35) yields

a E ≤ i * (a) E + 2ε.
Hence the desired inequality a E ≤ i * (a) E by letting ε go to zero.

Functorial properties of the volume entropy semi-norm

In this section, we present functorial properties of the volume entropy semi-norm and observe similarities with the ones satisfied by the simplicial volume.

Theorem 3.1.

(1) Let f : X → Y be a continuous map between two path-connected topological spaces. Then for every a ∈ H m (X; R)

f * (a) E ≤ a E .
(2) Let f : M → K(π 1 (M ), 1) be the classifying map of an orientable connected closed manifold M . Then

f * ([M ]) E = M E .
(3) Let f : M → N be a degree d map between two oriented connected closed manifolds.

Then

M E ≥ |d| N E .
(4) Let f : M → N be a d-sheeted covering map between two orientable connected closed manifolds.

Then

M E = d N E .
(5) Let M 1 and M 2 be two orientable connected closed manifolds of dimension m ≥ 3. Then

M 1 M 2 E ≤ M 1 E + M 2 E . (3.1)
(6) Let M be an orientable connected closed m-manifold with a negatively curved locally symmetric metric g 0 . Then

M E = Ω(M, g 0 ).
In particular, if M is a closed genus g surface then

M E = π M ∆ = 4π(g -1).
Remark 3.2. The properties (1)-( 5) are also satisfied by the simplicial volume. However, the simplicial volume is additive under connected sum in dimension at least 3; see [START_REF] Gromov | Volume and bounded cohomology[END_REF]. That is, there is equality in (3.1) if one replaces the volume entropy semi-norm with the simplicial volume. This leads to the following questions. Is there equality in (3.1)? Similarly, is there equality in (2.22)? A difficulty to overcome is that, contrarily to the simplicial volume, there is no cohomological interpretation of the volume entropy semi-norm.

Remark 3.3. It follows from (3) that both the simplicial volume and the volume entropy seminorm of an orientable connected closed manifold admitting a map to itself of degree different from 0 and ±1 are equal to zero. In a different direction, by Theorem 4.12, both the simplicial volume and the volume entropy semi-norm do not vanish for orientable connected closed manifolds admitting a negatively curved Riemannian metric; see [START_REF] Gromov | Volume and bounded cohomology[END_REF].

Proof of Theorem 3.1.

(1) Observe that if (M, ϕ) is a geometric cycle representing a ∈ H m (X; R) then (M, f • ϕ) is a geometric cycle representing f * (a) ∈ H m (Y ; R). Moreover, ω ker f * (M ) ≥ ω ker(ϕ•f ) * (M ) since ker f * ker(ϕ • f ) * . This immediately implies (1).

(2) For m = 2, the assertion is obvious since the classifying map is the identity map when M = S 2 . In case M = S 2 , all the terms of the relation vanish.

Suppose that m ≥ 3. The inequality f * ([M ]) E ≤ M E follows from [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF]. For the reverse inequality, consider the manifold M k = M • • • M defined as the connected sum of k copies of M and the degree k map f k : M k → M contracting all attaching spheres into a point. By definition,

Ω ker (f k ) * (M k ) ≥ Ω(k [M ]). (3.2)
The composite map

F k = f • f k : M k → K(π 1 (M ), 1) represents the class k f * ([M ]) ∈ H m (π 1 (M ); Z), that is, (F k ) * ([M k ]) = k f * ([M ]).
Observe also that it is π 1 -surjective. By [13, Theorem 10.2], this implies that

Ω(k f * ([M ])) = Ω ker (F k ) * (M k ). (3.3) Since f is π 1 -injective, Ω ker (F k ) * (M k ) = Ω ker (f k ) * (M k ).
Combined with (3.2) and (3.3), we derive

Ω(k f * ([M ])) k ≥ Ω(k [M ]) k .
By letting k go to infinity, we obtain

f * ([M ]) E ≥ M E , which implies (2).
(3) By definition, the assertion (3) immediately follows from (1).

(4) Let (Q, ψ) be a geometric cycle representing the class k [N ] ∈ H m (N ; Z). By adding handles to Q and mapping them to a generating set of π 1 (N ) if necessary (contracting the meridian spheres of the handles to points), we can assume that the map ψ : Q → N is π 1surjective. By Theorem 2.8 (and Remark 2.9 when m = 2), adding such handles does not increase the (relative) minimal volume entropy of the geometric cycle. Now, denote by P → Q the covering of Q corresponding to the subgroup (ψ * ) -1 (Im f * ). Note that P is an oriented connected closed m-pseudomanifold and that the covering map P → Q is of degree d. The map ψ : Q → N lifts to a map ϕ : P → M such that the following diagram

P M Q N ϕ f ψ commutes. Observe that the geometric cycle (P, ϕ) represents the class k [M ] ∈ H m (M ; Z).
Fix a piecewise Riemannian metric on Q and lift it to P . Since P → Q is a d-sheeted covering map and ker ϕ * = ker ψ * , we have Ω ker ϕ * (P ) ≤ d Ω ker ψ * (Q). Thus,

Ω(k [M ]) ≤ d Ω(k [N ]).
Dividing this inequality by k and letting k go to infinity, we obtain

M E ≤ d N E .
The reverse inequality M E ≥ d N E follows from (3).

(5) The idea is to realize [M 1 M 2 ] as the sum of [M 1 ] and [M 2 ] in a common topological space and to apply Corolllary 2.11. Let K i = K(π 1 (M i ), 1) be a classifying space for M i . Since m ≥ 3, the bouquet K = K 1 ∨ K 2 is a classifying space for M 1 M 2 . By the Mayer-Vietoris theorem, we have

H m (K; Z) H m (K 1 ; Z) ⊕ H m (K 2 ; Z).
Denote by [M i ] K ∈ H m (K; Z) the image of the fundamental class of M i under the homology homomorphism induced by the map f K i : M i → K, which is defined as the composite of the classifying map f i : M i → K i and the inclusion map [START_REF] Babenko | Topologie des systoles unidimensionnelles[END_REF] and the triangle inequality of the volume entropy semi-norm, see Corollary 2.11, we derive

K i → K = K 1 ∨ K 2 . Observe that f * ([M 1 M 2 ]) = [M 1 ] K + [M 2 ] K where f : M 1 M 2 → K is the classifying map of M 1 M 2 . By
M 1 M 2 E = f * ([M 1 M 2 ]) E ≤ [M 1 ] K E + [M 2 ] K E .

By (1), we also have [M

i ] K E ≤ [M i ] E = M i E .
Hence the result. [START_REF] Babenko | Minimal volume entropy and fiber growth[END_REF] The proof proceeds from a mild improvement on the minimal volume entropy estimate for closed manifolds admitting nonzero maps onto closed negatively curved locally symmetric manifolds; see [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]. This mild improvement, leading to (3.6), was carried out in [START_REF] Sambusetti | Minimal entropy and simplicial volume[END_REF]Theorem 2.5] for n ≥ 3 with the construction of a nonexpanding-volume map following [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]. Our approach is similar, except that it rests on the calibration argument of [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] (which can be applied to pseudomanifolds) and applies to both cases n ≥ 3 and n = 2. We refer to [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] for the notations (accordingly renaming M to X), the definitions and further details. Y e -c d g (y,z) dv g (z)

1 2
where p 0 is the Poisson kernel of ( X, g0 ); see [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]. The arguments of [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF] show that the map Ψ c is only defined when c > ent ker f * (Y, g) and that

det g (g Ψc ) ≤ c m (4m) m/2 (3.4)
where g Ψc is the pull-back under Ψ c of the Hilbert metric on L 2 (∂ X, dθ). Loosely speaking, the equivariant map Ψ c converges to the composite of f with the embedding √ p 0 of X into L 2 (∂ X, dθ) given by the Poisson kernel when c goes to ent ker f * (Y, g). The Poisson embedding √ p 0 is an isometry up to some factor (i.e., g √ p 0 = ent ker f * (Y,g) 4m

g 0 ) admitting an equivariant calibration form in the infinite dimensional sphere S ∞ + as explained below. Denote by π : S ∞ + → X the barycenter map, i.e., π(ρ) = bar(ρ 2 (θ) dθ); see [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]. The following calibration result was established in [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]Proposition 5.7] for m ≥ 3 and in [9, Theorem 6.2] for m = 2. Let ω 0 be the volume form on ( X, g0 ). The π 1 (X)-equivariant closed m-form π * ω 0 on S ∞ + calibrates the embedding Φ

0 : X → S ∞ + ⊂ L 2 (∂ X, dθ) defined as Φ 0 (x) = p 0 (x, •). Furthermore, comass(π * ω 0 ) = (4m) m/2 ent(X, g 0 ) m . (3.5)
Now, the map π • Ψ c is homotopic to π • Ψ 0 , where Ψ 0 = Φ 0 • f , through equivariant maps from Y to X. By (3.4) and (3.5), we derive

|(π • Ψ c ) * (ω 0 )| = |Ψ * c (π * ω 0 )| ≤ c ent(X, g 0 ) m |ω g |.
By a calibration argument, we deduce that

|deg(f )| • vol(X, g 0 ) ≤ Y |(π • Ψ c ) * (ω 0 )| ≤ c ent(X, g 0 ) m vol(Y, g)
passing the volume form to the quotient. As c go to ent ker f * (Y, g), we obtain

Ω(X, g 0 ) ≤ Ω ker f * (Y, g) k . (3.6)
Taking the infimum over all piecewise Riemannian metrics g on Y and over all geometric cycles (Y, f ) representing k [X], we derive Ω(X, g 0 ) ≤ X E after letting k go to infinity. The reverse inequality is obvious.

Remark 3.4. The assertion ( 6) can be extended to the following case. If M is an orientable connected closed 2m-manifolds given by a compact quotient of the product of m hyperbolic planes

H 2 then M E = Ω(M, g 0 )
where g 0 is the unique locally symmetric metric of minimal volume entropy on M among all locally symmetric metrics of given volume. Indeed, the proof can be adapted to follow the argument of [START_REF] Merlin | Minimal entropy for uniform lattices in product of hyperbolic planes[END_REF] based on the same calibration method as [START_REF] Besson | Entropies et rigidités des espaces localement symétriques de courbure strictement négative[END_REF]. Loosely speaking, we replace X by the m-product

H 2 × • • • × H 2 , ∂ X by the Furstenberg boundary ∂ F (H 2 × • • • × H 2 ) =
T m , dθ by the product Lebesgue probability measure on T m , and p 0 (x, θ) by the product p 0 (x 1 , θ 1 ) • • • p 0 (x m , θ m ) of the Poisson kernel of ∂H 2 . The main difference is that the calibration form π * ω 0 of the π 1 (X)-equivariant embedding Φ 0 is not constructed from the barycenter map π, but is given by a combinatorial (2m -1)-cocycle of T m ; see [START_REF] Merlin | Minimal entropy for uniform lattices in product of hyperbolic planes[END_REF] for the detail.

We need a couple of definitions to present the next result.

Definition 3.5. For i = 1, 2, let V i be a R-vector space endowed with a semi-norm . i . The tensor product V 1 ⊗ V 2 inherits the semi-norm . ⊗ given by the tensor product of . 1 and . 2 ; see [START_REF] Schaefer | Topological vector spaces[END_REF]. By definition, for every u ∈ V 1 ⊗ V 2 , we have

u ⊗ = inf s x s 1 y s 2 | u = s x s ⊗ y s (3.7)
where the infimum is taken over all the representations of u by finite sums of simple tensor products.

In the sequel, we endow the direct sum of semi-normed vector spaces with the direct sum of the semi-norms. With this convention, the graded vector space of the real homology H * (X; R) of a path-connected topological space X is endowed with the graded volume entropy semi-norm . * E given on each homogenous component by

a m E = 1 m m a E for every a ∈ H m (X; R).
The real homology of the direct product X 1 × X 2 of two path-connected topological spaces X 1 and X 2 is canonically endowed with two semi-norms. The first one is the usual volume entropy semi-norm . * E . The second one is defined via Künneth's formula

H m (X 1 × X 2 ; R) i+j=m H i (X 1 ; R) ⊗ H j (X 2 ; R)
as the tensor product norm, see (3.7), of the graded volume entropy semi-norms .

(i) * E on H * (X i ; R). It is denoted by . ⊗ E . We can now state our next result. Theorem 3.6. Let X 1 and X 2 be two path-connected topological spaces. Then, for every a ∈ H * (X 1 × X 2 ; R), the following inequality holds

a * E ≤ a ⊗ E . Proof.
It is enough to prove the inequality for homogeneous elements. Every homology class a ∈ H m (X 1 × X 2 ; R) admits a representation as a sum of simple tensor products

a = s x s ⊗ y s (3.8)
where x s ∈ H is (X 1 ; R) and y s ∈ H js (X 2 ; R) with i s + j s = m. By the triangle inequality,

a * E ≤ s x s ⊗ y s * E . (3.9)
By multiplying if necessary the homology class a by an appropriate natural number, we can suppose that all classes x s and y s in (3.8) are represented by closed manifolds. Proposition 2.6 of [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF] implies that x s ⊗ y s * E ≤ x s * E y s * E . Plugging this bound in (3.9) and minimizing over all the simple tensor product representations (3.8), we obtain the desired inequality.

Volume entropy semi-norm and simplicial volume

In this section, we show that the volume entropy semi-norm of a homology class is bounded from above and below by its simplicial volume, up to some multiplicative constants depending only on the dimension of the homology class. Therefore, the volume entropy semi-norm and the simplicial volume are equivalent homology semi-norms.

Geometrization of the simplicial volume.

Let us introduce some topological invariants. Definition 4.1. Let K be an m-dimensional topological space supplied with a finite pseudotriangulation (also referred to as a pseudo-simplicial complex or a ∆-complex ; see [29, §2.1]). Loosely speaking, the space K is a finite cell complex where the closure of each cell is homeomorphic to the standard simplex of the same dimension. In comparison with usual simplicial complexes, a simplex in a pseudo-triangulation is not uniquely defined by its vertices. An mdimensional geometric ∆-cycle is a disjoint finite union of m-dimensional ∆-complexes whose pseudo-triangulations satisfy the conditions (1), ( 2 Note that the function κ(n a) is subadditive in n, which ensures the existence of the limit (4.1). Furthermore, κ ∞ (a) ≤ κ(n a) n for every n ≥ 1.

We will also need the following definition extension the notion of simplicial volume to homology classes with coefficients in more general rings. Definition 4.2. Let X be a topological space and A = Z or Q. For every a ∈ H m (X; A), define

a A ∆ = inf c c 1
where the infimum is taken over all m-cycles c ∈ C(X; A) with coefficients in A representing a.

We present a couple of known results, including the proofs for the sake of completeness. The first result can be found in [START_REF] Schmidt | L 2 -Betti numbers of R-spaces and the integral foliated simplicial[END_REF]Lemma 2.9]. See [27, 5.41.(a)] for a previous statement.

Lemma 4.3. Every homology class a ∈ H m (X; Z) satisfies a ∆ = a Q ∆ .
Proof. Let σ (resp. σ ) be a real (resp. rational) m-cycle representing the real homology class induced by a. The difference σ -σ is the boundary of an (m + 1)-chain c ∈ C m+1 (X, R), that is, σ -σ = ∂c. By density of Q in R, there is a rational (m + 1)-chain c (with the same support) such that c -c 1 is arbitrarily small. Since the boundary of every (m + 1)-simplex is formed of m + 2 simplices of dimension m, we have

∂z 1 ≤ (m + 2) z 1 for every (m + 1)-chain z ∈ C m+1 (X; R). Thus, σ -(σ + ∂c ) 1 ≤ ∂(c -c ) 1 ≤ (m + 2) c -c 1
is arbitrarily small. Therefore, the real cycle σ and the rational cycle σ + ∂c , which both represent the real homology class induced by a, have arbitrarily closed • 1 -semi-norms. Hence the result.

Proposition 4.4. Every homology class a ∈ H m (X; Z) satisfies a Z ∆ = κ(a) (4.2)
a ∆ = κ ∞ (a). (4.3)
Proof. The inequality a Z ∆ ≤ κ(a) is obvious and the reverse inequality κ(a) ≤ a Z ∆ can be found in [35, Proposition 2.1] (and also follows from [29, p. 108-109]). Hence the relation (4.2).

Applying the average procedure of (4.1) to the obvious inequality a ∆ ≤ κ(a) yields the bound a ∆ ≤ κ ∞ (a). For every ε > 0, we also have

κ(n a) n = n a Z ∆ n ≤ a Q ∆ + ε
for some positive integer n, where the first equality follows from (4.2). Thus, κ ∞ (a) ≤ a Q ∆ + ε by subadditivity of the function κ(n a) with respect to n. By Lemma 4.3, this yields the bound κ ∞ (a) ≤ a ∆ + ε for every ε > 0. Hence the relation (4.3).

Universal realization of homology classes.

Let us introduce some results in geometric topology following [START_REF] Gaifullin | Realization of cycles by aspherical manifolds[END_REF], [START_REF] Gaifullin | The manifold of isospectral symmetric tridiagonal matrices and the realization of cycles by aspherical manifolds[END_REF] and [START_REF] Gaifullin | Universal realisators for homology classes[END_REF], which rely on C. Tomei's work [START_REF] Tomei | The topology of isospectral manifolds of tridiagonal matrices[END_REF].

Definition 4.5. The m-permutahedron Π m is the convex hull of the (m + 1)! points obtained by permutations of the coordinates of the point (1, 2, . . . , m + 1) of R m+1 . It is an m-dimensional simple convex polytope of R m+1 with 2 m+1 -2 facets, i.e., (m -1)-faces, that lies in the hyperplane

x 1 + • • • + x m+1 = m+1 j=1 j = (m + 1)(m + 2) 2 .
Here, an m-polytope is simple if each of its vertices is contained in exactly m facets. From a more geometric point of view, see [START_REF] Gaifullin | Combinatorial realisation of cycles and small covers[END_REF], the m-permutahedron Π m can be obtained by truncating the standard simplex ∆ m ⊆ R m+1 given by

x 1 + • • • + x m+1 = 1
with x i ≥ 0 as follows. First, truncate the vertices of ∆ m by the hyperplanes x i = 1 -1 4 . Then, truncate the edges of ∆ m by the hyperplanes

x i 1 + x i 2 = 1 - 1 4 2 .
At the k-th step, truncate the (k -1)-faces of ∆ m by the hyperplanes

x i 1 + • • • + x i k = 1 - 1 4 k .
The resulting polytope is combinatorially equivalent to the m-permutahedron Π m . The faces F of Π m correspond to the faces ∆ = ∆ F of ∆ m after truncation of which they appear. Consider the canonical piecewise linear map Θ : Π m → ∆ m which takes every face F of Π m to its corresponding face ∆ F in ∆ m . More precisely, define Θ on the barycenters b F of the faces of T m by sending b F to the barycenter of ∆ F . Then extends this map linearly to every simplex of the barycentric subdivision of T m ; see [START_REF] Gaifullin | Universal realisators for homology classes[END_REF]. Note that Θ is a degree one map which is injective in the interior of Π m . Definition 4.6. Consider the manifold M 0 of real symmetric tridiagonal matrices of size m + 1 with eigenvalues λ i = i, for i = 1, . . . , m + 1. Here, a matrix A = (a i,j ) is tridiagonal if a i,j = 0 whenever |i -j| > 1. The manifold M 0 will be referred to as the isospectral m-manifold.

It was proved by C. Tomei [START_REF] Tomei | The topology of isospectral manifolds of tridiagonal matrices[END_REF] that the isospectral manifold M 0 is an orientable closed aspherical m-manifold. By [START_REF] Tomei | The topology of isospectral manifolds of tridiagonal matrices[END_REF], [START_REF] Davis | Some aspherical manifolds[END_REF], [START_REF] Gaifullin | Realization of cycles by aspherical manifolds[END_REF], [START_REF] Gaifullin | The manifold of isospectral symmetric tridiagonal matrices and the realization of cycles by aspherical manifolds[END_REF] and [START_REF] Gaifullin | Universal realisators for homology classes[END_REF], the isospectral m-manifold M 0 is tiled by 2 m copies of the m-permutahedron Π m . More precisely, the manifold M 0 can be decomposed as M 0 (Z m 2 × Π m )/∼ where the equivalence relation is generated by (s, x) ∼ (r |ω| s, x) whenever x ∈ F ω . Here, the elements r i are the standard generators of Z m 2 . We will rely on the following universal property established by A. Gaifullin [START_REF] Gaifullin | Realization of cycles by aspherical manifolds[END_REF], [START_REF] Gaifullin | The manifold of isospectral symmetric tridiagonal matrices and the realization of cycles by aspherical manifolds[END_REF] regarding Steenrod's problem and the realization of cycles by closed manifolds. Theorem 4.7 ([20], [START_REF] Gaifullin | The manifold of isospectral symmetric tridiagonal matrices and the realization of cycles by aspherical manifolds[END_REF]). Let X be a path-connected topological space. Then for every homology class a ∈ H m (X; Z), there exist a connected finite-fold covering M 0 → M 0 of the isospectral manifold and a map f :

M 0 → X such that f * ([ M 0 ]) = q a
for some positive integer q depending on a. Remark 4.8. When m = 2, the isospectral surface M 0 is the genus 2 surface. More precisely, the permutahedron is an hexagon, the surface M 0 is tiled with 4 copies of this hexagon and these 4 copies surround every vertex of M 0 ; see [START_REF] Tomei | The topology of isospectral manifolds of tridiagonal matrices[END_REF]. Let us state the main theorem of this section. Theorem 4.9. Let m be a positive integer. Then there exists a constant C m > 0 such that every homology class a ∈ H m (X; Z) of a path-connected topological space X satisfies

a E ≤ C m a ∆ .
Proof. Let a ∈ H m (X; Z). By Proposition 4.4, see (4.3), for every ε > 0 and every integer s large enough, there exists a map h = h s : P → X from an m-dimensional geometric ∆-cycle

P = P s such that h * ([P ]) = s a (4.4) s a ∆ ≤ κ(P ) ≤ s( a ∆ + ε). (4.5)
The second barycentric subdivision of P gives rise to a simplicial structure on P ; see [START_REF] Hatcher | Algebraic topology[END_REF]. In general, the complex P is not connected. After the second barycentric subdivision, we can take the connected sum of the connected components by omitting out some m-simplices and gluing together the components to obtain an orientable connected closed pseudomanifold, still denoted by P . Note that this operation does not increase the number of m-simplices. Taking a third barycentric subdivision ensures that the simplicial structure admits a regular coloring in m + 1 colors (that is, any two vertices connected by an edge are of distinct colors) in order to apply some constructions of [START_REF] Gaifullin | Universal realisators for homology classes[END_REF]. Recall that the barycentric subdivision of a simplicial complex admits a regular coloring where every vertex which is the barycenter of an r-simplex of the original triangulation is of color r. The pseudomanifold P with this simplicial structure is denoted by Z. Since the barycentric subdivision of an m-simplex gives rise to m! simplices of dimension m, we obtain κ(Z) ≤ (m!) 3 κ(P ). (

By Theorem 4.7, there exists a map f :

M 0 → Z from a finite covering M 0 of M 0 such that f * ([ M 0 ]) = q [Z] ∈ H m (Z; Z) (4.7)
for some positive integer q.

Consider the piecewise flat metric on M 0 where all permutahedra are isometric to the standard permutahedron Π m with its canonical Euclidean metric. The volume of M 0 is equal to 2 m v m , where v m is the Euclidean volume of Π m .

By construction, see [START_REF] Gaifullin | Realization of cycles by aspherical manifolds[END_REF], [START_REF] Gaifullin | The manifold of isospectral symmetric tridiagonal matrices and the realization of cycles by aspherical manifolds[END_REF] and [START_REF] Gaifullin | Universal realisators for homology classes[END_REF], the map f : M 0 → Z satisfies the following features. The map f : M 0 → Z takes every permutahedron Π m of M 0 to a simplex ∆ m of Z and its restriction to Π m agrees with the canonical piecewise linear map Θ : Π m → ∆ m introduced in Definition 4.5. Furthermore, the number of permutahedra of the covering M 0 is equal to q κ(Z), where q is the degree of f ; see the end of the proof of Proposition 5.3 of [START_REF] Gaifullin | Universal realisators for homology classes[END_REF]. Therefore, the volume of M 0 satisfies vol(

M 0 ) = q κ(Z) v m (4.8)
where v m is the Euclidean volume of Π m .

Consider the composite map ϕ = h • f : M 0 → Z P → X. We derive from (4.7) and (4.4) that

ϕ * ([ M 0 ]) = qs a.
By definition of the volume entropy semi-norm, we have

qs a E ≤ ent ϕ ( M 0 ) m vol( M 0 ).
It follows from (4.8), (4.6) and (4.5) that

vol( M 0 ) ≤ q (m!) 3 s( a ∆ + ε) v m . Since ent ϕ ( M 0 ) ≤ ent(M 0 ), we deduce that qs a E ≤ qs (m!) 3 C m ( a ∆ + ε)
where C m = ent(M 0 ) m v m is a constant which only depends on m. Simplifying by qs and letting ε go to zero, we obtain

a E ≤ C m a ∆ where C m = (m!) 3 C m .
Remark 4.10. An estimate on the volume entropy ent(M 0 ) of the isospectral m-manifold provides an estimate on the constant C m in Theorem 4.9. Let us show a reverse inequality to Theorem 4.9.

Theorem 4.12. Let m be a positive integer. Then there exists a constant c m > 0 such that every homology class a ∈ H m (X; Z) of a path-connected topological space X satisfies

a E ≥ c m a ∆ .
In order to prove this theorem, we will need the following classical interpretation of the simplicial volume in terms of bounded cohomology; see [START_REF] Gromov | Volume and bounded cohomology[END_REF] for the definitions. Proposition 4.13. Let X be a topological space. Then every homology class a ∈ H m (X; R) satisfies

a ∆ = sup 1 α ∞ | α ∈ H m b (X; R), α, a = 1
where H m b (X; R) denotes the bounded cohomology of X of degree m. The following result is a technical extension of M. Gromov's inequality (1.6).

Proposition 4.14. Let m be a positive integer. Then there exists a constant c m > 0 such that for every map Φ : M → X from an oriented connected closed m-manifold M to a path-connected topological space X, we have

Ω ker Φ * (M ) ≥ c m Φ * ([M ]) ∆ .
S (x) = e -Rdḡ(x,•) -e -λR 1 Bḡ(x,R) (•) dvol ḡ(•).
By [24, §2.5], see also [START_REF] Balacheff | Macroscopic Schoen conjecture for manifolds with nonzero simplicial volume[END_REF] for further details, there exists a positive constant A m depending only on m such that [S ] ≤ A m λ (4.10) for R large enough. Technically speaking, the bound is stated in [START_REF] Gromov | Volume and bounded cohomology[END_REF] and [START_REF] Balacheff | Macroscopic Schoen conjecture for manifolds with nonzero simplicial volume[END_REF] when M is the universal covering of M , but the proof is exactly the same for intermediate coverings.

Integrating ω on M using the relation ω, [M ] = 1 and the combination of (4.9) with the bounds β ∞ ≤ α ∞ and (4.10), we obtain We can now proceed to the proof of Theorem 4.12.

1 ≤ m! (A m ) m α ∞ ent ker Φ * (M ) m vol(M ).
Proof of Theorem 4.12. For every ε > 0, there exists a positive integer k such that

Ω(k a) ≤ k ( a E + ε).
Thus, there exists a map ϕ : P → X defined on an oriented connected closed m-pseudomanifold P such that ϕ * ([P ]) = k a and Ω ker ϕ * (P ) ≤ Ω(k a) + ε ≤ k a E + (k + 1)ε. Remark 4.15. By a density argument, it follows from Theorem 4.9 and Theorem 4.12 that the volume entropy semi-norm and the simplicial volume semi-norm are equivalent in real homology, and not only in integral homology A natural question would be to determine whether or not the two semi-norms are proportional or not in every degree, though we do not have any strong evidence for that.

Systolic volume of a multiple homology class

The following asymptotically optimal upper bound on the systolic volume of the multiples of a given homology class positively answers a conjecture of [START_REF] Babenko | Systolic volume of homology classes[END_REF], where a sublinear upper bound was established. The systolic growth of finitely generated linear groups has been described by K. Bou-Rabee and Y. Cornulier; see [START_REF] Bou-Rabee | Systolic growth of linear groups[END_REF]. Originally stated in terms of residual girth rather than in terms of systolic growth, their result can be written as follows. We need to review some features of the isospectral m-manifold M 0 introduced in Definition 4.6.

It was proved by C. Tomei [START_REF] Tomei | The topology of isospectral manifolds of tridiagonal matrices[END_REF] that M 0 is an orientable closed aspherical m-manifold. By M. Davis [START_REF] Davis | Some aspherical manifolds[END_REF], its fundamental group G = π 1 (M 0 ) is isomorphic to a torsion-free subgroup of finite index of the Coxeter group W = s 1 , . . . , s m , r 1 , . . . , r m | s 2 i = r 2 i = 1, s i s j = s j s i for |i -j| > 1, s i s i+1 s i = s i+1 s i s i+1 , r i r j = r j r i , s i r j = r j s i for i = j .

Recall that J. Tits showed that every Coxeter group admits a faithful linear representation into a finite-dimensional vector space; see [12, Chap. V, §4, Corollary 2]. Thus, the group G is linear. This is an important feature in view of Theorem 5.3.

We can now proceed to the proof of Theorem 5.1.

Proof of Theorem 5.1. Let G = π 1 (M 0 ). Fix a finite symmetric generating set S of G once and for all. The metric on M 0 induced by the Hilbert-Schmidt metric (also called the Frobenius metric) on the space M m+1 (R) of square matrices of size m + 1 lifts to a metric d 0 on the universal covering M 0 of M 0 . (Here, the choice of the metric does not matter. We simply fix one once and for all.) Since M 0 is compact, its universal covering M 0 is quasi-isometric to (G, d S ); see [28, IV.B, Theorem 23]. More precisely, there exist some constants A 0 > 1 and B 0 > 0 such that for every γ ∈ G and every x ∈ M 0 , we have

A -1 0 d S (e, γ) -B 0 ≤ d 0 (x, γ • x) ≤ A 0 d S (e, γ) + B 0 . (5.1)
Note that A 0 and B 0 only depend on m. By Theorem 4.7, there exist a map f : M 0 → X from a finite covering M 0 of M 0 and a positive integer q such that

f * ([ M 0 ]) = q [a].
(5.2)

Let Γ G := π 1 ( M 0 ) be a finite index subgroup of G. Denote by f Γ : M 0 /Γ → X the lift of f : M 0 → X under the canonical projection π Γ : M 0 /Γ → M 0 . By the first inequality of (5.1), we have

A -1 0 sys(Γ, d S ) -B 0 ≤ sys( M 0 /Γ) ≤ sys f Γ ( M 0 /Γ). (5.3)
Now, apply Theorem 5.3 about the systolic growth of linear groups to the finitely generated linear group G. Thus, there exists a sequence of subgroups Γ

k G of finite index [ G : Γ k ] = k ≥ 2 such that sys(Γ k , d S ) ≥ C 0 log k (5.4)
for some C 0 > 0 which does not depend on k.

Let M k = M 0 /Γ k . We derive from (5.3) that sys f k ( M k ) ≥ A -1 0 C 0 log k -B 0 ≥ D 0 log k (5.5)
where f k : M k → X is the lift of f : M 0 → X and D 0 > 0 does not depend on k. Since Γ k is of index k in G and the map f : M 0 → X represents q a, we deduce that

(f k ) * ([ M k ]) = kq a. Since vol( M k ) = k vol( M 0 ), this yields the inequalities σ(kq a) ≤ σ f k ( M k ) ≤ C k (log k) m
for every k ≥ 2, where C = C (a) does not depend on k. Since σ is sub-additive, see [5, Proposition 3.6], we derive

σ(k a) ≤ σ k q q a ≤ C k (log k) m
for every k ≥ 2, where C = C(a) does not depend on k.

Systolic volume semi-norm and functorial properties

In this section, we define the systolic volume semi-norm and present its functorial properties along with some comparison results.

6.1. Systolic volume semi-norm.

Theorem 5.1 allows us to define the systolic volume semi-norm in real homology of dimension m ≥ 3. This definition is based on the following observation, whose proof is left to the reader. Lemma 6.1. Let M be a Z-module endowed with a translation-invariant pseudo-distance . Given a function h : N → R + with lim k→∞ h(k) = ∞, suppose that for every a ∈ M, there is a

positive constant C = C(a) such that (0, k a) ≤ C h(k) for every k ∈ N. Then (a, b) = lim sup k→∞ (k a, k b) h(k) (6.1)
defines a translation-invariant pseudo-distance on M.

Let X be a path-connected topological space. Apply Lemma 6.1 with h(k) = where σ(a) = (0, a). Remark 6.2. The behavior of σ(k a) as a function of k can be quite irregular; see [5, §5.4]. This suggests it may not be possible to replace the lim sup in (6.1) by the usual limit for (a, b) = σ(a -b). It is also unclear, though unlikely, whether the second stablization process in (6.2) can be omitted in the definition of the systolic volume semi-norm.

We following lemma is useful to establish upper bounds on the systolic volume semi-norm. As in Section 3, we establish functorial properties of the systolic volume semi-norm similar to the ones satisfied by the simplicial volume. Theorem 6.4. Let m ≥ 3 be an integer.

(1) Let f : X → Y be a continuous map between two path-connected topological spaces. Then for every a ∈ H m (X; R) f * (a) σ ≤ a σ .

(2) Let f : M → K(π 1 (M ), 1) be the classifying map of an orientable connected closed manifold M . Then f * ([M ]) σ = M σ .

(3) Let f : M → N be a degree d map between two oriented connected closed manifolds. Then M σ ≥ |d| N σ .

(4) Let f : M → N be a d-sheeted covering map between two orientable connected closed manifolds. Then M σ = d N σ .

(5) Let M 1 and M 2 be two orientable connected closed manifolds of dimension m ≥ 3. Then

M 1 M 2 σ ≤ M 1 σ + M 2 σ . (6.3)
Remark 6.5. As in Remark 3.2, we can ask whether equality holds in (6.3). Here is a consideration suggesting this question might be rather subtle. By Theorem A of [START_REF] Babenko | Géométrie systolique des sommes connexes et des revêtements cycliques[END_REF], for every essential m-manifold M 1 with m ≥ 4, there exists an essential m-manifold M 2 such that

σ(M 1 M 2 ) < σ(M 1 ) + σ(M 2 ).
Now, it is unclear whether this strict inequality subsists or not under the double stabilization process in the definition of the systolic volume semi-norm. More generally, the inequalities (3.1), (6.3) and the one for the simplicial volume reflect convexity properties of the semi-norms. Even if the three semi-norms are equivalent (as we will see), we have no strong evidence that the volume entropy semi-norm and the systolic volume semi-norm satisfy the same additivity property as the simplicial volume.

Proof of Theorem 6.4. We argue as in the proof of Theorem 3.1, pointing out only the differences.

(

) Observe that if (M, ϕ) is a geometric cycle representing a ∈ H m (X; R) then (M, f • ϕ) is a geometric cycle representing f * (a) ∈ H m (Y ; R). Since every (f • ϕ)-noncontractible loop of M is f -noncontractible, we derive sys f (M ) ≤ sys f •ϕ (M ). Hence, σ f (M ) ≥ σ f •ϕ (M ) 1
. This implies that σ(f * (a)) ≤ σ(a). Replacing a by k a and applying Lemma 6.3.(1), we obtain the desired inequality f * (a) σ ≤ a σ .

(2) We consider the degree k map f

k : M k → M defined on the connected sum M k = M . . . M of k copies of M . By definition, σ f k (M k ) ≥ σ(k [M ]). Consider also the composite map F k = f • f k : M k → K(π 1 (M ), 1), where (F k ) * ([M k ]) = k f * ([M ]). Note that F k is π 1 -surjective. By [13, Theorem 10.2], this implies that σ F k (M k ) = σ(k f * ([M ])). Since f is π 1 -injective, sys F k (M k ) = sys f k (M k ). Hence, σ F k (M k ) = σ f k (M k ). Combining the previous estimates, we obtain σ(k f * ([M ])) ≥ σ(k [M ]).
By Lemma 6.3.(1), we derive f * ([M ]) σ ≥ M σ . Since the reverse inequality follows from (1), this implies (2).

(3) By definition, the assertion (3) immediately follows from (1).

( (5) The proof of similar to the proof of the point (5) in Theorem 3.1. Simply replace the volume entropy semi-norm • E by the systolic volume semi-norm • σ .

) Construct a geometric cycle (Q, ψ) representing k [N ] ∈ H m (N ; Z) with ψ : Q → N π 1 - surjective, whose (relative) systolic volume σ ψ (Q) is arbitrarily close to σ(k [N ]). Consider the geometric cycle (P, ϕ) representing k [M ] ∈ H m (M ; Z) where P is the cover of Q corresponding to the subgroup (ψ * ) -1 (Im f * ) and ϕ : P → M is the lift of ψ : Q → N . Since P → Q is a d-sheeted 4

Systolic volume comparison.

We present analogues of comparison results obtained for the minimal volume entropy (seminorm) in Section 2 to the systolic volume (semi-norm) case.

It is convenient to introduce the following definitions. Let M be a connected closed m-manifold with a Riemannian metric g. Let H ¡ π 1 (M ) be a normal subgroup. Define the systole of M relative to H, denoted by sys H (M, g), as the length of the shortest loop of M whose homotopy class does not lie in H. As in (1.7), define the systolic volume of M relative to H as σ H (M ) = inf g vol(M, g) sys H (M, g) m where the infimum is taken over all (piecewise) Riemannian metrics g on M . This is a slight modification of the definition (1.7). For f : M → X and H = ker f * , the invariants σ f and σ H coincide. Note that the definition of σ H extends to finite simplicial complexes. Though Proposition 3.6 of [START_REF] Babenko | Systolic volume of homology classes[END_REF] is stated with the absolute systolic volume, its short proof based on the comparison principle [2, Proposition 3.2] can easily be adapted to cover the relative case. Thus, we obtain Proposition 6.6. For i = 1, 2, let M i be a connected pseudomanifold of dimension m ≥ 3 and M i ¡ π 1 (M i ) be a normal subgroup. Then

σ H 1 * H 2 (M 1 M 2 ) ≤ σ H 1 (M 1 ) + σ H 2 (M 2 ).
Theorem 2.12 holds true if one replaces the relative minimal entropy to the power m, namely Ω H , with the relative systolic volume σ H . (A previous version valid for "admissible" pseudomanifolds can be found in [START_REF] Babenko | Systolic volume of homology classes[END_REF]Corollary 3.5].) More precisely, we have Theorem 6.7. For i = 1, 2, let M i be a connected closed pseudomanifold of dimension m ≥ 3 and H i ¡ π 1 (M i ) be a normal subgroup. Let H ¡ π 1 (M 1 ) * π 1 (M 2 ) be a normal subgroup such that the canonical inclusion π 1 (M 1 ) π 1 (M 1 ) * π 1 (M 2 ) induces an inclusion π 1 (M 1 )/H 1 (π 1 (M 1 ) * π 1 (M 2 ))/H. Suppose M 2 is orientable. Then

σ H 1 (M 1 ) ≤ σ H (M 1 M 2 ).
Proof. The proof is similar to the one of Theorem 2.12, except that the bound on the volume entropy ent H 1 (M 1 , g ε ) at the end of the proof of Theorem 2.12 should be replaced with

sys H 1 (M 1 , g ε ) ≥ sys H 1 (M 1 M 2 ∪ M 2 (m-2)
Cone(M 2 (m -2)), g ) ≥ sys H (M 1 M 2 , g).

As previously, combining Theorem 6.7 and Proposition 6.6, we obtain the following analogue of Corollary 2.13 for the (relative) systolic volume. In the case of "admissible" pseudomanifolds, it follows from Corollary 3.5 and Proposition 3.6 of [START_REF] Babenko | Systolic volume of homology classes[END_REF]. Corollary 6.8. For i = 1, 2, let M i be a connected closed pseudomanifold of dimension m ≥ 3 and H i ¡ π 1 (M i ) be a normal subgroup. Suppose M 2 is orientable and σ H 2 (M 2 ) = 0. Then

σ H 1 * H 2 (M 1 M 2 ) = σ H 1 (M 1 ).
The analogue of Theorem 2.19 for the systolic volume semi-norm holds true. Note however that its proof differs from the one of Theorem 2.19. This is due to the fact that the systolic volume and the minimal volume entropy do not have the same behavior under finite coverings. Theorem 6.9. Let G be a finitely presented group and H be a finite index subgroup of G. Let m ≥ 3. Suppose that the canonical inclusion i : H → G induces a monomorphism between the m-dimensional rational homology groups

(i * ) m : H m (H; Q) → H m (G; Q).
Then for every homology class a ∈ H m (H; Z)

a σ = i * (a) σ .
Proof. Let a ∈ H m (H; Z). By Thom's theorem [START_REF] Thom | Quelques propriétés globales des variétés différentiables[END_REF], there exists an integer q ≥ 1 such that q i * (a) ∈ H m (G; Z) can be represented by a geometric cycle f : M → K(G, 1) where M is a closed m-manifold. By adding handles if necessary, we can further assume that f * : π 1 (M ) → G is surjective. Arguing as in the proof of Theorem 2.

Systolic volume semi-norm and simplicial volume semi-norm

In this section, we show that the systolic volume semi-norm and the simplicial volume seminorm are equivalent in real homology.

Let us show that the systolic volume semi-norm bounds from above the simplicial volume semi-norm and/or the volume entropy semi-norm (up to a multiplicative constant). Recall that every homology class a ∈ H m (X; Z) with m ≥ 3, where X is a path-connected topological space, satisfies σ(a) ≥ λ m a ∆ (log(2 + a ∆ )) m where λ m is a positive constant depending only on m; see (1.9). Therefore,

(log k) m k σ(k a) ≥ λ m a ∆ log k log(2 + k a ∆ ) m .
Letting k go to infinity, we obtain σ(a) ≥ λ m a ∆ (arguing separately whether a ∆ is zero or not). Thus, a σ ≥ λ m a ∆ .

Alternatively, every homology class a ∈ H m (X; Z) satisfies σ(a) ≥ λ m Ω(a) (log(2 + Ω(a))) m for some constant λ m > 0 depending only on m; see [START_REF] Sabourau | Systolic volume and minimal entropy of aspherical manifolds[END_REF] and [START_REF] Brunnbauer | Homological invariance for asymptotic invariants and systolic inequalities[END_REF]. Therefore, Again, letting k go to infinity, we obtain σ(a) ≥ λ m a E (arguing separately whether Ω(a) is zero or not). Thus, a σ ≥ λ m a E .

We establish a reverse inequality in the following theorem. Theorem 7.1. Let m ≥ 3 be an integer. Then there exists a constant µ m > 0 such that every homology class a ∈ H m (X; Z) of a path-connected topological space X satisfies a σ ≤ µ m a ∆ .

Proof. We argue as in the proof of Theorem 4.9 and Theorem 5.1. Let a ∈ H m (X; Z). For every ε > 0 and every integer s large enough, we consider the map h : Z → X from a closed connected simplicial m-pseudomanifold Z to X such that h * ([Z]) = s a κ(Z) ≤ s (m!) 3 ( a ∆ + ε) obtained at the beginning of the proof of Theorem 4.9. By Theorem 4.7, there exists a map f : M 0 → Z from a finite covering M 0 of M 0 such that

f * ([ M 0 ]) = q [Z] ∈ H m (Z; Z)
for some positive integer q. Applying Theorem 5.3 as in (5.4)

  This estimate allows us to define the systolic volume semi-norm in real homology of dimension m ≥ 3 as follows. By [5, Corollary 5.3], the systolic volume induces a translation-invariant pseudo-distance on H m (X; Z) with m ≥ 3, defined by (a, b) = σ(a -b). Define a new translation-invariant pseudo-distance on H m (X; Z) given by (a, b) = lim sup k→∞ (log k) m k (k a, k b). See Lemma 6.1 for further detail. Denote by σ(a) = (0, a)

  [START_REF] Babenko | Asymptotic invariants of smooth manifolds[END_REF] and f 2 by first collapsing the attaching sphere to a point. Note that ker f * ker(f 1 ) * * ker(f 2 ) * .

  H k (M (k), M (k -1); Z) as free abelian groups with basis the k-cells e k α of M , the cellular boundary formula, see [29, §2.2], gives ∂(e m ) = s deg(φ s ) e m-1 s where φ s : S m-1 → M (m -1) → S m-1 s is the composite of the attaching map φ of the mcell e m = D m with the quotient map collapsing M (m -1) \ e m-1 s to a point. (Note that φ s factorizes through φ.) Since ∂ = 0, every map φ s is contractible. Hence φ is null-homotopic as desired.

Proposition 2 . 16 .

 216 Let M be an oriented connected closed manifold of dimension m ≥ 3 and f : M → K(π 1 (M ), 1) be its classifying map. Suppose f * ([M ]) ∈ H m (π 1 (M ); Z) is a finite order homology class. Then Ω(M ) = 0. Proof. Let N = M . . . M be the manifold obtained by taking the connected sum of k copies of M . Consider the map F = f ∨• • •∨f : N → K(π 1 (M ), 1) obtained by collapsing each attaching sphere to a point and by applying f to each term M in the bouquet M ∨ • • • ∨ M . The class F * ([N ]) is equal to k a, where a = f * ([M ]). Suppose k a = 0. Since N is an oriented connected closed manifold and the homomorphism F * : π 1 (N ) → π 1 (M ) induced by F is surjective, we derive from [13, Theorem 10.2], see (1.4), that Ω ker F * (N ) = 0. Apply Theorem 2.12 to M 1 = M and the connected sum M 2 = M . . . M of k -1 copies of M by taking H = ker F * . This immediately leads to the desired result.

π 1 -

 1 surjective and to take a lift corresponding to the subgroup H G. The resulting geometric cycle is almost extremal and represents dka up to some torsion element, where d = [G : H]. Denote by p the number of generators in G and by d = [G : H] the index of H on G. Let

  Let f : Y → X = M be a map from an oriented connected closed m-pseudomanifold representing k [X]. The map f lifts to a map f : Y → X, where Y is the covering of Y with π 1 (Y ) = ker f * . (This is the main difference with [9, §8], where the map is lifted to Ỹ → X.) Given a piecewise Riemannian metric g on Y , denote by g the lifted metric on Y . Fix c > 0. Consider the π 1 (Y )/ ker f * -equivariant map Ψ c : Y → S ∞ + ⊂ L 2 (∂ X, dθ) defined as Ψ c (y, θ) = Y e -c d g (y,z) p 0 (f (z), θ) dv g (z)

  ) and (3) of Definition 2.1. The geometric complexity of K, denoted by κ(K), is the number of m-simplices of K. Define the geometric complexity of a homology class a ∈ H m (X; Z) as κ(a) = inf P κ(P ) where P runs over the m-dimensional geometric ∆-cycle representing a. That is, there is a map h : P → X such that h * ([P ]) = a, where the class [P ] is the sum of the fundamental classes of the connected components of P with the appropriate orientations. Define also the average geometric complexity of a as κ ∞ (a) = lim n→∞

4. 3 .

 3 Homology norm comparison: upper bound on the volume entropy semi-norm.

Remark 4 . 4 . 4 .

 444 [START_REF] Bou-Rabee | Systolic growth of linear groups[END_REF]. A referee pointed out to us that Theorem 4.9 can be derived from [15, Proposition 7.11] combined with Theorem 3.1.(4), once the notion of the volume entropy semi-norm is well established. The current presentation allows us to make a connection with the proof of Theorem 5.1. Homology norm comparison: lower bound on the volume entropy semi-norm.

Proof.

  Fix a Riemannian metric g on M . Let M → M be the covering of M with fundamental group ker Φ * . The quotient group Γ = π 1 ( M )/ ker Φ * acts by deck transformations on M . Denote by M( M ) the Banach space of finite (signed) measures µ on M with the norm µ = M |µ|. Denote also by M + ( M ) ⊆ M( M ) the cone of positive measures. Following [24, §2.4], a smoothing operator on M is a smooth Γ-equivariant map S : M → M + ( M ). Define [S ] = sup x∈ M d x S S (x).Let α ∈ H m b (X; R) such that α, Φ * ([M ]) = 1, where •, • is the the bilinear pairing between cohomology and homology given by the Kronecker product. Defineβ = Φ * (α) ∈ H m b (M ; R). Clearly, β ∞ ≤ α ∞ and β, [M ] = 1. By[START_REF] Gromov | Volume and bounded cohomology[END_REF] Proposition, p.33], there exists a closed m-form ω on M representing the cohomology classβ ∈ H m (M ; R) such that ω ≤ m! β ∞ [S ] m (4.9)for every smoothing operator S : M → M + ( M ). For λ > ent ker Φ * (M ), define S = S λ,R : M → M + ( M ) as

  Hence,c m Φ * ([M ]) ∆ ≤ Ω ker Φ * (M ) by Proposition 4.13, with c m = (m! (A m ) m ) -1, where A m is the multiplicative constant in (4.10).

(4. 11 )

 11 By Thom's theorem[START_REF] Thom | Quelques propriétés globales des variétés différentiables[END_REF], there exists a map f : M → P defined on an oriented connected closed m-manifold M such thatf * ([M ]) = d [P ] ∈ H m (P ; Z)for some suitable nonzero integer d. Extend f : M → P by handle attachements into a π 1surjective map f : M → P whereM = M l i=1 S 1 × S m-1 . Clearly, f * ([M ]) = f * ([M ]) = d [P ].By [1, Proposition 2.2], we have Ω ker Φ * (M ) ≤ d Ω ker ϕ * (P ) (4.12) where Φ : M → X is the composite map Φ = ϕ • f . Now observe that Φ * ([M ]) = dk a. By Proposition 4.14, we derive c m dk a ∆ ≤ Ω ker Φ * (M ). (4.13) Combining the inequalities (4.11), (4.12) and (4.13), dividing by dk and letting ε go to zero, we obtain c m a ∆ ≤ a E as desired.

Theorem 5 . 1 .

 51 Let m be a positive integer. For every homology class a ∈ H m (X; Z) of a pathconnected topological space X, there exists a constant C = C(a) > 0 such that for every k ≥ 2, we haveσ(k a) ≤ C k (log k) m .The proof of Theorem 5.1 rests on some systolic estimates in geometric group theory based on the following notion. Definition 5.2. Let G be a finitely generated group and S be a finite generating set of G. Denote by d S the word distance induced by S. For every finite index subgroup Γ G, define sys(Γ, d S ) = inf γ∈Γ\{e} d S (e, γ).

Theorem 5 . 3 .

 53 Let G be a finitely generated linear group over a field and S be a finite symmetric generating set of G. Then there exist a constant C 0 > 0 and an infinite sequence of subgroupsΓ k G of finite index k such that sys(Γ k , d S ) ≥ C 0 log k.Remark 5.4. A similar estimate has been previously stated without proof by M. Gromov for finitely generated subgroups G of SL d (Z) under the extra assumption that no unipotent element lies in G; see [26, Elementary Lemma, p. 334].

k(

  log k) m (see Theorem 5.1) to the translation-invariant pseudo-distance defined on H m (X; Z) with m ≥ 3 by (a, b) = σ(a -b); see [5, Corollary 5.3]. This yields a new translation-invariant pseudodistance on H m (X; Z). Define the systolic volume semi-norm of a ∈ H m (X; Z) as a σ = lim k→∞ σ(k a) k (6.2)

Lemma 6 . 3 .

 63 Let a, b ∈ H m (X; Z) with m ≥ 3, where X is a path-connected topological space.

( 1 )( 1 ) 6 . 2 .

 1162 Suppose that there exists λ ≥ 0 such that σ(k a) ≤ λ σ(k b) for every integer k ≥ 1. Then a σ ≤ λ b σ . (2) Suppose that there exists σ 0 ≥ 0 such that σ(k a) ≤ k (log k) m σ 0 for every integer k ≥ 2. Then a σ ≤ σ 0 . Proof. By assumption, replacing k by kp, we have σ(kp a) ≤ λ σ(kp b). Dividing this inequality by k (log k) m and letting k go to infinity, we obtain σ(p a) ≤ λ σ(p b). Dividing this inequality by p and letting p go to infinity, we derive the desired bound a σ ≤ λ b σ . (2) By assumption, replacing k by kp, we have σ(kp a) ≤ kp (log k + log p) m σ 0 . Dividing this inequality by k (log k) m and letting k go to infinity, we derive σ(p a) ≤ p σ 0 . Dividing this inequality by p and letting p go to infinity, we obtain a σ ≤ σ 0 as desired. Functorial properties of the systolic volume semi-norm.

  covering map and ker ϕ * = ker ψ * , we have σ ϕ (P ) ≤ d σ ψ (Q). Thus, σ(k [M ]) ≤ d σ(k [N ]). By Lemma 6.3.(1), we derive the inequality M σ ≤ d N σ . The reverse inequality follows from (3).

  19, we construct a commutative diagram M K(H, 1) M K(G, 1) f f where the vertical maps are d-sheeted coverings, such that f * ([M ]) = q i * (a) and f * ([ M ]) = dq a + c, where c ∈ TorH m (H; Z).Denote by t the order of the torsion class c. For every integer k ≥ 1, consider the connected sum M k of kt copies of M and the map f k : M k → K(G, 1) obtained by collapsing the attaching spheres of the connected sum M k to a point and by applying f : M → K(G, 1) to each term M of the bouquet so-obtained. By construction, the mapf k : M k → K(G, 1) is π 1 -surjective with (f k ) * ([M k ]) = kt f * ([M ]) = ktq i * (a). Since M k is a closed manifold and f k : M k → K(G, 1) is π 1 -surjective, we derive from [2, 3] or [13, Theorem 10.2] that σ f k (M k ) = σ(ktq i * (a)).The liftf k : M k → K(H, 1) of f k : M k → K(G, 1) to the corresponding covers M k of M k represents ( f k ) * ([ M k ]) = kt f ([ M ]) = kt (dq a + c) = ktdq a. Since the vertical map M k → M k is a d-sheeted covering, we have σ f k ( M k ) ≤ d σ f k (M f ). Hence, σ(ktdq a) ≤ σ f k ( M k ) ≤ d σ f k (M k ) = d σ(ktq i * (a))for every k ≥ 1. By Lemma 6.3.(1) and the homogeneity of the systolic volume semi-norm, we deduce that a σ ≤ i * (a) σ . The reverse inequality i * (a) σ ≤ a σ follows from Theorem 6.4.(1).

(

  log k) m k σ(k a) ≥ λ m Ω(k a) k log k log(2 + k Ω(a)) m .

3 D m 0 v m . Remark 7 . 2 .

 372 yields a sequence of subgroups Γ k in the finitely generated linear groupG := π 1 ( M 0 ) with finite index [ G : Γ k ] = k ≥ 2 such that sys(Γ k , d S ) ≥ C 0 log kfor some positive constant C 0 (which does not depend on k), by applying Theorem 5.3 as in(5.4). Denote byf k : M k → Z the lift of f : M 0 → Z to the cover M k = M 0 /Γ k corresponding to the subgroup Γ k G. Since Γ k is of index k in G, we derive (f k ) * ([ M k ]) = kq [Z]. Define ϕ k = h • f k : M k → Z → X. Observe that (ϕ k ) * ([ M k ]) = kqs a.(7.1)As in (5.5), we havesys ϕ k ( M k ) ≥ sys f k ( M k ) ≥ D 0 log k (7.2)where D 0 is a positive constant (which does not depend on k or a). As in (4.8), we also havevol( M k ) = k vol( M 0 ) = kq κ(Z) v m ≤ kqs (m!) 3 ( a ∆ + ε) v m . (7.3)It follows from (7.1), (7.2) and (7.3) that σ(qks a) ≤ σ ϕ k ( M k ) ≤ kqs (m!) 3 D m 0 (log k) m ( a ∆ + ε) v m By Lemma 6.3.(2), this implies qs a σ ≤ qs (m!) 3 D m 0 ( a ∆ + ε) v m .Hence, a σ ≤ µ n a ∆ , where µ n = (m!) By a density argument, the systolic volume semi-norm and the simplicial volume semi-norm are equivalent in real homology, and not only in integral homology.

  [START_REF] Davis | Some aspherical manifolds[END_REF], the volume entropy semi-norm of M (and its simplicial volume) vanishes. Proposition 2.16 asserts that the volume entropy of M also vanishes, without any stabilization process.Example 2.18. Every manifold M with fundamental group SL(2, Z) Z 4 * Z 2 Z 6 or P SL(2, Z) Z 2 * Z 3 has zero minimal volume entropy, that is, Ω(M ) = 0. Indeed, the homology of an amalgamated product can be computed through a Mayer-Vietoris sequence involving the homology groups of its factors. Since the homology of every cyclic group is composed of finite groups

(except in dimension zero), the same holds for the homology groups of SL(2, Z) and P SL(2, Z); see [33, Theorem 4.1.1].

2.6. Volume entropy semi-norm comparison.

Let us give an application of Corollary 2.13. Theorem 2.19. Let G be a finitely presented group and H be a finite index subgroup of G. Let m ≥ 3. Suppose that the canonical inclusion i : H → G induces a monomorphism between the m-dimensional rational homology groups

Partially supported by the ANR project Min-Max (ANR-19-CE40-0014).

Density of the volume entropy and systolic volume semi-norms of manifolds

The following density result can be deduced from the equivalence of the semi-norms given by Theorem 1.3 and Theorem 1.6 with the recent work [START_REF] Heuer | The spectrum of simplicial volume[END_REF]. M k = M • • • M as the connected sum of k copies of M . By additivity of the simplicial volume under connected sums in dimension at least three, see [START_REF] Gromov | Volume and bounded cohomology[END_REF], we have M k ∆ = k M ∆ . It follows from the equivalence of the semi-norms, see Theorem 1.3 and Theorem 1.6, that the sequence M k * starts from the interval (0, ε), with 0 < M * < ε, and goes to infinity. Now, the map M k+1 → M k collapsing one copy of M to a point is of degree 1. By the functorial properties of the volume entropy semi-norm, namely (3) and ( 5) of Theorem 3.1 and Theorem 6.4, we derive

Thus, the sequence M k * is nondecreasing and increases by at most M * < ε at each step. We deduce from the properties of the sequence M k * that every interval of [0, ∞) of length ε contains at least one term M k * . Hence the desired density result.