Alexander Kolpakov

Leone Slavich

Hyperbolic 4-manifolds, colourings and mutations

Nous proposons une nouvelle approche à construire des variétés hyperboliques M en dimension quatre, par moyens d'un polyèdre de Coxeter P ⊂ H 4 munit avec un coloriage de ses faces. Aussi, nous utilisons notre méthode pour obtenir des sous-surfaces totalement géodésiques plongées dans M, et pour décrire le résultat de mutations par rapport à ses surfaces. Comme application, nous construisons une variété complète hyperbolique non-compacte X avec un bout cuspide non-torique, et une variété complète hyperbolique non-compacte Y avec un bout cuspide torique. Elles sont des nouveaux exemples de variétés hyperboliques complètes non-compactes en dimension quatre avec un seul bout cuspide et volume relativement petit.

Introduction

By Margulis' Lemma, a finite-volume complete hyperbolic n-manifold M n has a finite number of ends called cusps, each of which is homeomorphic to N n-1 ×[0, +∞) for some closed Euclidean (n-1)-manifold N n-1 . In dimension three many knot complements provide an example of a hyperbolic manifold with a single cusp [START_REF] Thurston | Geometry and Topology of 3-Manifolds[END_REF]. In this setting the resulting manifolds are necessarily orientable, and since the torus T 2 is the only example of a closed Euclidean surface, the cusp is homeomorphic to T 2 × [0, +∞).

In higher dimensions, where there is no direct analogue to geometrisation, constructing hyperbolic manifolds is a harder task. The goal is usually achieved by using either arithmetic methods, or producing direct constructions based on Poincaré's fundamental polytope theorem.

It is also known that the cusp section of a single-cusped hyperbolic 4-manifold cannot have any possible homeomorphism type of a closed Euclidean 3-manifold [START_REF] Long | On the geometric boundaries of hyperbolic 4-manifolds[END_REF]. In fact, a necessary condition for such a manifold to be the cusp section of a single-cusped hyperbolic 4-manifold is that its η-invariant be an integer, and such a condition is met by only four out of the six possible orientable homeomorphism types. The first example of a single-cusped hyperbolic 4-manifold whose cusp section is a 3-torus was given in [START_REF] Kolpakov | Hyperbolic four-manifold with one cusp[END_REF]. To the best of our knowledge, no example of single cusped hyperbolic 4-manifold with cusp section non-homeomorphic to the 3-torus has been constructed before. In this paper, we provide an example of such an object (Theorem 1.1). The cusp section of our example is homeomorphic to the unique Euclidean orientable S 1 -bundle over the Klein bottle K (see [START_REF] Scott | The geometries of 3-manifolds[END_REF]).

It would also be interesting to generalise to dimension four the well-known 3-dimensional examples of hyperbolic knot complements. A reasonable model would be the complement of a 2-dimensional torus T in S 4 . The boundary ∂(U (T )) of a regular neighbourhood U (T ) of T will necessarily be homeomorphic to the 3-torus T 3 . Therefore ∂(U (T )) will admit a Euclidean structure, and we can hope to find a complete, finite volume hyperbolic structure on the complement of S where ∂(U (T )) corresponds to a cross-section of the cusp.

However we have to be careful: since a manifold M of dimension n ≥ 4 can admit a variety of non-equivalent smooth structures, there is a sharp difference in saying that M is homeomorphic or diffeomorphic to some preferred model of a smooth manifold. First examples of hyperbolic 4-manifolds which are homeomorphic to complements of collections of 2-dimensional tori in S 4 were given in [START_REF] Ivanšić | Complements of tori and Klein bottles in the 4-sphere that have hyperbolic structure[END_REF]. All the examples constructed there have more than one cusp, and each of these cusps is homeomorphic to T 3 ×[0, +∞), where T 3 denotes the three-dimensional torus. In [START_REF] Saratchandran | A four dimensional hyperbolic link complement in a standard S 4[END_REF], one of these manifolds (equipped with the smooth structure induced by its hyperbolic metric) is shown to be diffeomorphic to the complement of a collection of five tori in S 4 with its standard smooth structure.

For a hyperbolic manifold M , being the complement of a collection of tori or Klein bottles in S 4 implies strong restrictions on the geometry and topology of M itself. A classical result due to H.C. Wang states that, in all dimensions n ≥ 2, the volumes of finite-volume hyperbolic n-manifolds form a well-ordered subset of R, which is discrete if n = 3 [START_REF] Vinberg | Geometry. II: Spaces of constant curvature[END_REF]Theorem 3.1]. Moreover, in even dimensions, the volume of a hyperbolic n-manifold M turns out to be proportional to the Euler characteristic χ(M) through the Gauß-Bonnet formula ( [START_REF] Serre | Cohomologie des groupes discrets[END_REF], see also [START_REF] Heckman | The volume of hyperbolic Coxeter polytopes of even dimension[END_REF][START_REF] Zehrt | The covolume of discrete subgroups of Iso(H 2m )[END_REF])

Vol M = c n • χ(M ), with c n = (-2π) n 2 /(n -1)!!.
The Euler characteristic of a four-dimensional hyperbolic manifold M (in general, non-compact, though finite-volume) can take any positive integer value [START_REF] Ratcliffe | The volume spectrum of hyperbolic 4-manifolds[END_REF]. However, if such a manifold is the complement of a collection of tori in S 4 , an easy excision property argument shows that necessarily χ(M ) = χ(S 4 ) = 2, and therefore Vol M = 8π 2 /3.

In this setting, combining several properties of a manifold in a single example may become a difficult task (e.g. we want to construct a complete hyperbolic finite-volume 4-manifold with a single cusp, and a small volume). In the present paper we use the technique of colourings (see [START_REF] Davis | Convex polytopes, Coxeter orbifolds and torus actions[END_REF][START_REF] Garrison | Small covers of the dodecahedron and the 120-cell[END_REF][START_REF] Kolpakov | Some hyperbolic three-manifolds that bound geometrically[END_REF][START_REF] Yu | Vesnin: Three-dimensional hyperbolic manifolds of Löbell type[END_REF][START_REF]Three-dimensional hyperbolic manifolds with a common fundamental polyhedron[END_REF]) in order to produce examples of single-cusped hyperbolic manifolds of Euler characteristic 2. The main results are the following: Theorem 1.1. There exists an orientable, complete, finite-volume hyperbolic 4-manifold X such that χ(X ) = 2 with a single cusp whose cross-section is homeomorphic to the Euclidean orientable S 1 -bundle over the Klein bottle.

Theorem 1.2. There exists an orientable, complete, finite-volume hyperbolic 4-manifold Y such that χ(Y) = 2 with a single cusp whose cross-section is homeomorphic to a 3-torus.

We point out the fact that the examples constructed in this work have minimal volume amongst known examples of orientable, single cusped, hyperbolic 4-manifolds. Finally, we point out that the techniques introduced in this paper allow us to produce many non-compact hyperbolic 4-manifolds of Euler characteristic 2. It is reasonable to wonder if some of these are new examples of complements of tori and/or Klein bottles in S 4 . Also, our examples of hyperbolic manifolds with toric cusps may be used to produce new instances of Einstein manifolds, c.f. [START_REF] Bamler | Construction of Einstein metrics by generalised Dehn filling[END_REF], or integer homology spheres with non-vanishing simplicial volume, c.f. [START_REF] Fujiwara | Simplicial volume and fillings of hyperbolic manifolds[END_REF][START_REF] Ratcliffe | Some examples of aspherical 4-manifolds that are homology 4-spheres[END_REF].

The paper is organised as follows: in Section 2, we recall basic properties of colourings and prove some auxiliary statements. Then, in Section 3, we classify all possible colouring of a cube, up to homeomorphism of the resulting manifolds. In Section 4 we study the properties of colourings of a particular polytope: the 4-dimensional Potyagaȋlo-Vinberg polytope first introduced in [START_REF] Potyagaȋlo | On right-angled reflection groups in hyperbolic space[END_REF]. In Section 5 we produce a colouring of P 4 that we use in order to construct our examples. Finally, in Section 6, we use mutations along 3-dimensional hyper-surfaces in 4-dimensional multi-cusped hyperbolic 4-manifolds to construct singlecusped hyperbolic 4-manifolds with toric and non-toric cusp sections.

Properties of colourings

In this section we review the construction of manifolds from colourings of right-angled polytopes, with particular attention to the case of non-compact hyperbolic polytopes. We also discuss the relations between the algebraic properties of the colouring and the topological properties of the resulting manifold.

Let X n with X = S, E, or H denote, respectively, the n-sphere, n-dimensional Euclidean space, or n-dimensional hyperbolic (or Lobachevskiȋ) space. Let P ⊂ X n be a convex polytope, and let F(P ) be the set of its co-dimension one faces (or facets). A polytope P ⊂ X n is called simple if each of its vertices belongs to exactly n facets.

A colouring of a simple polytope P according to [START_REF] Davis | Convex polytopes, Coxeter orbifolds and torus actions[END_REF][START_REF] Garrison | Small covers of the dodecahedron and the 120-cell[END_REF][START_REF] Yu | Vesnin: Three-dimensional hyperbolic manifolds of Löbell type[END_REF][START_REF]Three-dimensional hyperbolic manifolds with a common fundamental polyhedron[END_REF] is a map λ : F(P ) → (Z/2Z) n . A colouring is called proper if the colours of facets around each vertex of P are linearly independent vectors of V = (Z/2Z) n .

One of most remarkable cases is a colouring of a compact right-angled polytope P ⊂ H n , with a proper colouring1 . Polytopes of this type give rise to interesting families of hyperbolic manifolds [START_REF] Garrison | Small covers of the dodecahedron and the 120-cell[END_REF][START_REF] Kolpakov | Some hyperbolic three-manifolds that bound geometrically[END_REF][START_REF] Yu | Vesnin: Three-dimensional hyperbolic manifolds of Löbell type[END_REF][START_REF]Three-dimensional hyperbolic manifolds with a common fundamental polyhedron[END_REF].

In [START_REF] Kolpakov | Some hyperbolic three-manifolds that bound geometrically[END_REF] the notion of a colouring is extended to let V be any finite-dimensional vector space over Z/2Z. Below we shall use the notion of colouring in a wider context of right-angled polytopes P , which are not simple. A polytope P ⊂ X n is called simple at edges if each edge belongs to exactly (n -1) facets. In the case of a right-angled polytope P ⊂ H 4 , this means that vertex figures of P are either 3-dimensional tetrahedra or cubes [START_REF] Dufour | Notes on right-angled Coxeter polyhedra in hyperbolic spaces[END_REF]Proposition 1].

A colouring of a polytope P ⊂ X n which is simple at edges is a map λ : F(P ) → V , where V is a finite-dimensional vector space over Z/2Z. A colouring λ is proper if the following two conditions are satisfied:

1. Properness at vertices: if v is a simple vertex of P , then the n colours of facets around it are linearly independent vectors of V ;

2. Properness at edges: if e is an ideal edge (i.e. an edge having an ideal vertex, or both ideal vertices) of P , then the (n -1) colours of facets around e are linearly independent.

Let P be a right-angled polytope in X n , with X = S, E, or H. In the latter case, we allow some (not necessarily all) of the vertices of P to lie on the ideal boundary2 . Let λ : F(P ) → V be a colouring on the facets of P , with values in a finite-dimensional vector space V over Z/2Z, which satisfies all the properness conditions.

The colouring defines a homomorphism from G to V which we continue to denote by λ, where G = G(P ) is the right-angled Coxeter group generated by reflections in the facets of P . The following key result holds: Proposition 2.1. Let λ be a coloring of a right-angled polytope which satisfies all the properness conditions at hyperbolic vertices and at the ideal edges. The kernel of the associated homomorphism λ : G → V is a torsion-free subgroup of G. To this subgroup, there corresponds an orbifold cover

π : M λ → P,
where M λ is a manifold.

Proof. Consider the tessellation of H n obtained by reflecting the polytope P in its facets. The group G acts on H n preserving such tessellation. A torsion element in G necessarily fixes a face F of the tessellation. Up to conjugation, we can suppose that F is a face of P . The stabilizer of F is generated by the reflections in the facets of P which contain F . By the properness condition, we see that the stabilizer of F is mapped injectively in V by λ and the proposition follows.

Let us suppose that the homomorphism λ is surjective. The automorphism group of the covering is isomorphic to V . To visualise the automorphism, consider the manifold M λ , together with its tessellation into copies of the polytope P . Consider any copy of P 1 of P in the tessellation of M λ . The polytope P 1 will be adjacent to another copy P 2 of P along a facet F . The automorphism corresponding to λ(F ) permutes P 1 and P 2 by reflection in their common facet F .

Let Γ λ be a rooted graph defined as follows:

• the root of Γ λ is a vertex p corresponding to a copy of P , and labelled with λ(p) = 0 ∈ V ,

• if two copies P 1 and P 2 of P in the tessellation of M λ corresponding to the vertices p 1 and p 2 of Γ λ are adjacent through a facet F , then the edge {p 1 , p 2 } is labelled λ(F ),

• if two copies P 1 and P 2 of P in the tessellation of M λ are adjacent through a facet F , then the labels λ(p 1 ), λ(p 2 ) ∈ V of the respective vertices p 1 and p 2 of Γ λ satisfy λ(p 1 ) + λ(p 2 ) = λ(F ).

We call Γ λ the developing graph of λ. If a vertex p of Γ λ has label v, its root can be shifted into p by adding the vector v to all the vertex labels and declaring p the new root. Thus, the choice of the root and the vertex labelling is not canonical, while the choice of the edge labels is canonical. Therefore, the vertices of the graph Γ λ , can be interpreted as points of an affine space A over V . Now, let F be a facet of P . The facet F lifts to a finite collection of totally geodesic hyper-surfaces in M λ . To the facet F , there corresponds a subgroup W (F ) ⊂ V generated by λ(F ) and the colours λ(F ′ ) of all the facets F ′ of P adjacent to F . Proposition 2.2. The following properties hold:

(1) The number of totally geodesic hyper-surfaces in M λ corresponding to the facet F is equal to the index of W (F ) in V .

(2) Cutting along all the hyper-surfaces associated with the facets F 1 . . . , F n which share a common colour λ(F 1 ) = • • • = λ(F n ) separates M λ if and only if the colour λ(F i ) does not lie in the subspace U generated by all other colours.

(3) The hyper-surfaces associated with a facet F are two-sided if and only if the colour λ(F ) does not lie in the subspace generated by the colours of all the facets adjacent to F .

(4) If two hyper-surfaces H 1 and H 2 are lifts of two non-intersecting facets of P , then they do not intersect in the manifold M λ .

(5) The manifold M λ is orientable if and only if, for some isomorphism V ∼ = (Z/2Z) s , each colour λ(F i ) has an odd amount of 1's.

Proof. [START_REF] Bamler | Construction of Einstein metrics by generalised Dehn filling[END_REF] The number of lifts of a facet F is the order of the orbit of a single lift F of F to M λ under the action of the deck transformation group. It equals the index of Stab( F ), the stabiliser of F under this action. Obviously, Stab( F) = W (F ).

(2) Let M λ := M λ //{π -1 (F 1 ), . . . , π -1 (F n )} be the manifold M λ cut along all the lift of F i 's. Remove from the developing graph for the colouring λ all the edges with label λ(F i ) to obtain the developing graph Γ λ for the manifold M λ . It is clear that the graph Γ λ is connected if and only if the colour λ(F i ) lies in the subspace generated by all other colours, therefore the same property holds for the manifold M λ .

(3) Let F 1 , . . . , F k be the faces of P adjacent to a given facet F . Let us choose as a root for the developing graph Γ λ a vertex v adjacent to an edge with label λ(F ), and consider the connected subgraph Γ F λ ⊂ Γ λ given by the vertices of Γ λ which can be joined to the root v by edges with labels λ(F ), or λ(F i ), i = 1 . . . k.

The graph Γ F λ is the developing graph of a small tubular neighbourhood Tube(H F ) of a lift H F of a facet F in M λ , where each vertex corresponds to the intersection of a copy of the polytope P with the neighbourhood Tube(H F ). The hyper-surface H F is two-sided in M λ if and only if Tube(H F ) \ H F is disconnected. By applying an argument analogous to that of (2), we see that this holds if and only if λ(F ) does not lie in the subspace generated by the colours λ(F i ), i = 1 . . . k.

(4) Consider the manifold M λ , together with its tessellation into copies of the polytope P . The hyper-surfaces associated with a facet F of P are naturally tessellated by copies of the facet F . If two hyper-surfaces H 1 and H 2 intersect, they do so in the boundary of some copy of P , and therefore correspond to two facets F 1 , F 2 of P such that F 1 ∩ F 2 = ∅.

(5) See [START_REF] Kolpakov | Some hyperbolic three-manifolds that bound geometrically[END_REF]Lemma 2.4].

Definition 2.3. A coloring λ : F(P ) → V is orientable if it satisfies condition (5) of Proposition 2.2.

Equivalent colourings and coloured isometries

Given a colouring λ : G → V (from now on, we shall systematically denote the colouring and the associated homomorphism by the same letter), there are a number of equivalent colourings, that give rise to manifolds isometric to M λ . We can compose the homomorphism λ : G → V with any injective linear automorphism φ ∈ GL(V ) in order to obtain a new colouring, which we denote by φ * (λ). Formally speaking, we change each colour λ(F ) ∈ V associated with a facet F of P to the colour φ(λ(F )). It is clear that the kernel of the homomorphism φ • λ coincides with the kernel of λ, and therefore the manifolds M λ and M φ * (λ) are isometric. However this operation changes the correspondence between the group of automorphisms of the colouring, which is generated by reflections in the facets of P , and the vector space V .

Notice that every symmetry of the polytope P induces a natural permutation of its facet colours.

Definition 2.4. Let λ : G(P ) → V be a colouring. A symmetry ψ of P is admissible with respect to the colouring λ if the naturally associated map on the facet colours is realised by a linear automorphism φ ∈ GL(V ) of V . Admissible symmetries form a subgroup Adm λ (P ) of Symm(P ), and there is a naturally defined homomorphism from Adm λ (P ) to GL(V ). We denote by GL(λ) the image of this homomorphism. Now let Adm λ (P ) < Symm(P ) be the group of admissible symmetries of P with respect to a colouring λ : G(P ) → V . Each symmetry ψ ∈ Adm λ (P ) lifts to an isometry of M λ , and obviously any two lifts differ by composition with an element of Aut π ∼ = V . Definition 2.5. The group of lifts of admissible symmetries of P to M λ is called the coloured isometry group of M λ , and is denoted by Isom c (M λ ).

There is a short exact sequence

0 → Aut π ∼ = V → Isom c (M λ ) → Adm λ (P ) → 0.
This sequence naturally splits: it is sufficient to choose a copy P 0 of P in the tessellation of M λ , and lift every ψ ∈ Adm λ (P ) to the unique element of Isom c (M λ ) which maps P 0 to itself. Therefore we write

Isom c (M λ ) ∼ = V ⋊ Adm λ (P ),
where the action of Adm λ (P ) on V is induced by the natural action of GL(λ) on V .

Therefore, given a colouring λ : F(P ) → V , the group Isom c (M λ ) acts on the developing graph Γ λ through affine isomorphisms of its vertex set A. The elements of Aut π correspond to translations in A, while elements of Adm λ (P ) induce linear maps in GL(λ) of the underlying vector space V .

Moreover, to each stratum S in the cell decomposition of the manifold M λ , there corresponds an affine subspace W S ⊂ A. Notice that each such n-stratum S corresponds to a unique n-face F (S) of the polytope P . The elements of W S correspond to the copies of the polytope P which are adjacent to the stratum S. These form an affine subspace of A whose dimension is equal to the co-dimension of S. The underlying linear subspace of V is trivial in the case of strata of co-dimension 0 (copies of the polytope P ), and, for strata corresponding to a co-dimension one face F , it coincides with the subspace generated by the colour λ(F ). In all other cases, the underlying linear subspace coincides with the subspace generated by the colours of the facets of P adjacent to F (S).

Remark 2.6. In order to check if an element ψ ∈ Isom c (M λ ) acts on the manifold M λ without fixed points, it is sufficient to check if no affine subspace of the form W S ⊂ A is preserved by ψ.

Remark 2.7. Notice that the discussion above also applies to the cusps of M λ . Each of these will be associated with an ideal vertex v of the polytope P , and with each cusp we can associate an affine subspace of A, generated by the colours carried over to the vertex figure of v from the respective facets of P adjacent to v (i.e. a restriction of λ to the vertex figure of v).

Colourings of a three-dimensional cube

In this section we discuss an example of colouring on the faces of a 3-dimensional Euclidean cube, in order to show how to compute the topology and the geometry of the manifolds associated with such colourings. Moreover, we derive a criterion to determine the topology of a Euclidean manifold M λ associated with a colouring λ on the faces of a cube in terms of the intersections of the vector spaces generated by the colours on opposite faces.

Example 3.1. Here we exhibit an example of colouring λ on the faces of a unit cube C which produces a non-toric flat manifold M λ , namely, the one which which fibres over S 1 with a torus as fibre and monodromy given by a hyper-elliptic involution. This manifold can be described also as the Euclidean orientable S 1 -bundle over the Klein bottle K, and it is of topological type G 2 in Wolf's census of closed Euclidean 3-manifolds [START_REF] Wolf | Spaces of Constant Curvature[END_REF].

Consider a Euclidean unit cube with its three pairs of opposite square faces. We assign colours from V = (Z/2Z) 3 to them in the following way (see Figure 1):

1. Assign the colour (1, 0, 0) to a pair of opposite faces, 2. Assign the colour (0, 1, 0) to the second pair of opposite faces, 3. Assign the colours (0, 0, 1) and (1, 1, 1) to the remaining two faces.

The third pair of opposite faces is parallel to the fibres of the fibration. We may check that the monodromy is a hyper-elliptic involution by performing the following computation.

Let us start with a cube C 0 = C, labelled with 0 ∈ V and keep reflecting in its faces. We also translate the face colours from each copy of C to its reflected counterpart accordingly. If two copies We repeat this process until we are able to find three linearly independent isometries ψ i ∈ Isom E 3 , i = 1, 2, 3, which generate a manifold of the correct volume, namely the order, as a set, of the image of the colouring λ in the vector space V , since these guarantees that the isometries ψ i , i = 1, 2, 3, generate the automorphism group of the covering E 3 → M λ . In the case of the above described colouring of a cube (see Figure 1) we apply this process to obtain a sequence of five cubes depicted in Figure 2, where the first C 0 and the last C N = C 4 cubes are labelled 0 and the isometry identifying them is

ψ 1 ((x, y, z) T ) =   1 0 0 0 -1 0 0 0 -1     x y z   +   2 0 0  
Here, we start by reflecting the cubes in the (1, 1, 1)-face. Analogously, we obtain the respective isometries ψ 2 and ψ 3 by reflecting first in the (0, 1, 0)-or (1, 0, 0)-face. We get

ψ 2 ((x, y, z) T ) = (x, y + 2, z) T , ψ 3 ((x, y, z) T ) = (x, y, z + 2) T .
Then, the translation lattice of M λ is generated by ψ i 's, and its volume equals Vol M λ = 8. Thus, the manifold M λ has type G 2 according to [START_REF] Wolf | Spaces of Constant Curvature[END_REF].

The geometry and topology of the manifold M λ are determined only by the equivalence class of the colouring λ. Therefore, understanding the topology of the cusp sections starting from a colouring of a cube, depends only on the equivalence class of the colouring. This allows us to describe the topology of the resulting cusp shape by understanding the intersections between the different subspaces generated by the colours of pairs of opposite faces of that cube.

Let us denote by C the unit cube in the Euclidean 3-space E 3 = {(x, y, z) | x, y, z ∈ R}, and let G be the associated reflection group. The group G is generated by reflections in the planes of the form x = c, y = c, z = c, with c ∈ N, and is a right-angled Coxeter group. Reflections in parallel planes form three subgroups G x , G y , G z ⊂ G.

An orientable, proper V -colouring λ of the square faces of a unit cube C defines three linear subspaces of V , that we call V x , V y , V z ⊂ V , respectively, each generated by the colours assigned to a pair of opposite facets of C. The colouring also defines three homomorphisms

λ x : G x → V x , λ y : G y → V y and λ z : G z → V z .
In order to understand the topology of the manifold M λ obtained from the colouring λ of the cube C, we shall consider the restriction of λ to a section of C by the plane x = c, 0 < c < 1. This section is a unit square S, whose edges inherit their colours from the respective faces of C. Indeed, each edge of S arises at the intersection of the plane x = c with a face F of C, orthogonal to x = c.

The restriction of λ to S is an orientable colouring, with values in the vector space of the form V y + V z , and, by the orientability assumption, it produces a torus T x with volume 2 dim(Vy+Vz) . The vector space V y + V z acts on the torus T x by automorphisms of the covering T x → S. Clearly, all sections of C by the planes x = c, 0 < c < 1, lift to tori in M λ , which constitute the fibres of the fibration of M λ over the circle S 1 .

We note, that in the exposition below the variables x, y and z are interchangeable: no direction along which we may fibre M λ is preferred. The intersection properties of the subspaces V x , V y and V z that we are interested in remain intact under a permutation of x, y and z.

We shall determine the monodromy map T x → T x of the torus fibration over S 1 that represents M λ . Any orientation-preserving automorphism of T x is expressible as a sum of an even number of face colours assigned to the faces parallel to the planes y = 0 and z = 0, and is necessarily isotopic to the identity or to a hyper-elliptic involution. The elements of V y + V z that induce maps isotopic to the identity form a subgroup of V y + V z , which has index two in the group of orientation-preserving isometries of T x . It consists of elements which are expressed as a sum of an even number of face colours in both V y and V z .

The torus T x is obtained as a quotient of E 2 by translations along certain two vectors v 1 and v 2 . By embedding E 2 into E 3 = R × E 2 , we see that the vectors (0, v 1 ), (0, v 2 ) belong to the translation lattice associated with the Euclidean 3-manifold M λ . We can add a third translation in G x along a vector of the form v 3 = (n, 0), generated by reflections in the square faces parallel to x = 0, in order to define a lattice L. The volume of E 3 /L equals 2 dim (Vx)+dim (Vy+Vz) .

The length of the vector v 3 equals 2 if the subspace V x is one-dimensional (in other words, if the colours of the respective pair of opposite facets are the same), and it equals 4 if V x has dimension 2 (in this case, the colours of opposite faces corresponding to V x are distinct). Notice, that the lattice defined above does not necessarily coincide with the maximal translation lattice associated with the manifold M λ .

If the intersection V x ∩ (V y + V z ) is trivial, the lattice L is indeed the full group of isometries which defines the cover E 3 → M λ . To see this, it suffices to notice that

dim V = dim (V x + V y + V z ) = dim(V x ) + dim(V y + V z ).
This means that the torus defined by the lattice L has the same volume as the manifold M λ , and therefore the covering

E 3 /L → M λ is trivial. If the intersection V x ∩ (V y + V z
) is non-trivial, we have to consider all possible cases separately.

(1) Suppose that V x has dimension 1, and pick the unique vector w ∈ V x ∩ (V y + V z ). Clearly, the reflection in any of the two square faces of the cube C parallel to x = 0 gives an isometry ψ ∈ G x of E 3 such that λ x (ψ) = w.

Moreover, since w ∈ V y + V z = Aut(T x → S) we can find a (necessarily orientation-reversing) isometry φ of E 3 (belonging to the group generated by G y and G z ), such that λ x (φ) = w. The isometry φ induces an index two covering T x → K, where K is a Klein bottle. Moreover

φ • ψ ∈ ker λ = π 1 (M λ ).
The quotient of E 3 /L under this isometry is the manifold M λ , since

dim V = dim (V x + V y + V z ) = dim(V x ) + dim(V y + V z ) -1 and Vol (E 3 /L) = 2 • Vol (M λ ).
The resulting manifold can be described as the orientable Euclidean S 1 -bundle over the Klein bottle K, which is also a torus bundle over S 1 , with a hyper-elliptic involution acting on its fibres.

(2) Now, suppose that V x has dimension 2 (i.e. the colours of the faces of C parallel to x = 0 are different). There are two possible cases:

(2.a) If V x ∩ (V y + V z ) has dimension 1, it contains a unique non-zero vector w. If w is one of the face colours which generate V x , then we can proceed as in the previous case and verify that the resulting manifold is a torus bundle over the circle, with the hyper-elliptic involution as monodromy map.

If w is the sum of the face colours which generate V x , then the isometry ψ ∈ G x given by the translation along the vector (0, 0, 2) (which is generated by reflections in the faces of C that are parallel to x = 0), satisfies λ x (ψ) = w. As before, we can find an orientation-preserving isometry φ ∈ G y , G z of E 3 such that λ x (φ) = w. We need to check whether φ induces the identity or a hyper-elliptic involution on the torus T x , i.e. whether the vector w is the sum of an even or an odd number of the face colours in V x and V z . By proceeding as in the previous steps, we see that in the first case the resulting manifold is a torus. Otherwise, it is a torus bundle with a hyper-elliptic involution as monodromy.

(2.b) If V x ⊂ (V y + V z ), the resulting manifold is non-toric. We deduce, in analogy to [START_REF] Bamler | Construction of Einstein metrics by generalised Dehn filling[END_REF], that it is a torus bundle with a hyper-elliptic involution as monodromy.

We summarise the discussion above in the following way: Proposition 3.2. Let λ be a proper, orientable colouring on the faces of a Euclidean cube. The Euclidean manifold M λ is either a 3-torus, or a torus bundle over S 1 with monodromy given by a hyper-elliptic involution.

The manifold is a 3-torus if and only if one of the following two conditions in fulfilled:

1. One of the vector spaces V x , V y , V z ⊂ V has trivial intersection with the sum of the other two.

2. One of the vector spaces V x , V y , V z ⊂ V (without loss of generality, we can suppose that it is V x ), has a one-dimensional intersection with the sum of the other two, and the unique non-trivial vector w = w y + w z , w y ∈ V y , w z ∈ V z , in this intersection is expressed as the sum of the two face colours that generate V x , while the vectors w y and w z are expressed as the sum of an even number of each of the face colours that generate V y and V z , respectively.

4 Colourings of the polytope P 4

Here we shall discuss general properties of colourings of the right-angled hyperbolic polytope P 4 introduced by L. Potyagaȋlo and È. Vinberg in [START_REF] Potyagaȋlo | On right-angled reflection groups in hyperbolic space[END_REF], discuss an example and use it to build single cusped manifolds whose cusp section is either a 3-torus, or an orientable S 1 -bundle over the Klein bottle. We begin by describing the combinatorial dual of the polytope P 4 : the rectified 5-cell, which we denote by R. The polytope R is obtained by considering the convex hull in R 5 of the set of ten points obtained by permuting the coordinates of the vector (1, 1, 1, 0, 0). Alternatively, one can obtain the rectified 5-cell starting from a 4-dimensional Euclidean simplex S 4 , and considering the convex hull of the midpoints of its edges.

As a consequence of the latter construction for R, we see that there is the following sequence of one-to-one correspondences:

{Facets of P 4 } ↔ {Vertices of R} ↔ {Edges of S 4 } (1)
and defining a colouring on the facets of P 4 is equivalent to defining a colouring on the edges of a 4-simplex S 4 , i.e. an edge colouring of the complete graph K 5 .

The polytope P 4 has a hyperbolic realisation as a right-angled polytope with ten facets which are copies of the polyhedron P 3 [START_REF] Potyagaȋlo | On right-angled reflection groups in hyperbolic space[END_REF], depicted in Figure 3. Each copy is labelled as a vertex of the rectified 5-cell, i.e. by a permutation of the coordinates of the vector (1, 1, 1, 0, 0). Moreover, the polytope P 4 has 10 vertices in total. Of these, 5 are ideal and the other 5 are compact and lie inside the hyperbolic space. By marking the vertices of S 4 with the integers in the set {1, 2, 3, 4, 5}, each ideal (resp. compact) vertex of P 4 receives a natural label: opposite to each ideal vertex, there is a compact vertex with the same label. To an edge of S 4 connecting vertices with labels i and j, there corresponds a facet of P 4 whose label has 0's in the i-th and j-th entries, and 1's in all other entries. For instance, to the edge connecting the vertices 4 and 5 of S 4 , there corresponds a facet of P 4 with label (1, 1, 1, 0, 0).

The vertex figure of an ideal vertex is a Euclidean cube. The hyperbolic volume of P 4 is equal to (1/16) • v m [START_REF] Everitt | Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel[END_REF], where v m = 4π 2 /3 is the minimal volume of a hyperbolic 4-manifold.

Observe that each facet of P 4 corresponds to an edge of the complete graph K 5 , and that two edges of K 5 share a vertex if and only if the corresponding facets of P 4 are incident. Given an edge e of K 5 , there are exactly 3 other edges which do not share a vertex with e, and these edges form a complete sub-graph K 3 ⊂ K 5 . As a consequence we see that, given a facet F of P 4 , there are exactly 3 other (pairwise intersecting) facets of P 4 which are disjoint from F .

In order to obtain a manifold, we need to make sure that the colouring of P 4 is proper at both the compact vertices and the ideal edges. This is equivalent to the following two conditions imposed on the edge colouring of the graph K 5 :

Figure 3: The P 3 -facet of P 4 labelled (1, 1, 1, 0, 0), together with labels on its vertices. The vertices 1, 2, 3 are ideal, while 4 and 5 lie in hyperbolic space.

1. Properness at vertices: The colours associated with the edges emanating from a vertex v of K 5 are linearly independent vectors. There are 5 vertices to check.

2. Properness at edges: The colours associated with every complete sub-graph K 3 ⊂ K 5 (built by removing any two vertices and the edges incident to these vertices) are linearly independent vectors. There are 10 such sub-graphs to check.

Induced colourings

Now let F(P 4 ) be the set of facets of P 4 , and V a finite dimensional vector space over Z/2Z. Let us call M λ the manifold associated with a colouring λ : F(P 4 ) → V satisfying the properness conditions. Notice that, by the condition on the volume of P 4 , the dimension of V has to be at least 4. We want to study the induced colourings.

For each edge e of K 5 , we have a sub-graph of K 5 (called a dart) formed by the edge e and all other edges which share a vertex with e. It has 7 edges in total, 2 vertices of valence 4, and 3 vertices of valence 2, as shown in Figure 4. The edges of the dart associated with an edge e correspond to the facets of P 4 which are adjacent to the facet associated with e.

Given an edge e of K 5 , the colouring on its corresponding dart graph naturally induces a V F -colouring on the P 3 -facet F of P 4 corresponding to e, where V F = V / λ(F ) , and therefore defines an embedded, totally geodesic sub-manifold of the manifold M λ .

Each ideal vertex w of P 4 naturally corresponds to a sub-graph K 4 ⊂ K 5 . The colouring on the edges of K 4 , which correspond to the facets of P 4 that meet at w, naturally induces a colouring on the cubical vertex figure of w, with values in the vector space V w ⊂ V generated by the colours of the facets adjacent to w. The index of V w in V determines the number of cusps that correspond to the vertex w.

A highly symmetric example 5.1 Defining the colouring

Consider a colouring of the complete graph K 5 with values in V = (Z/2Z) 5 obtained in the following way:

1. Label the vertices of K 5 with the numbers {1, 2, 3, 4, 5}.

2. Assign to the edge connecting vertices i and j the element of V which has 0's in the entries i and j, and 1's in all other entries. For instance, the edge connecting vertices 1 and 2 will be coloured with (0, 0, 1, 1, 1).

This edge colouring of K 5 defines a facet colouring λ of the polytope P 4 . It is easy to check that the properness conditions are satisfied, and therefore this colouring defines an orientable hyperbolic fourmanifold M := M λ , tessellated by 2 5 = 32 copies of the polytope P 4 .

The group Adm λ (P 4 ) of admissible symmetries of P 4 coincides with its full isometry group Symm(P 4 ) ∼ = S 5 , where the action on V is given by permutation of the coordinates. Therefore we have a short exact sequence

0 → V → Isom c (M) → S 5 → 0 (2)

Identifying hyper-surfaces

By applying Proposition 2.2, we see that:

Remark 5.1. To each facet F of P 4 there corresponds a unique, non-separating, two-sided totally geodesic hyper-surface H F in M.

Moreover, since the group of admissible symmetries of the colouring acts transitively on the facets of P 4 , the hyper-surfaces defined by the induced colourings on the facets are isometric to each other.

The colouring of the dart graphs associated with the edges having colours (1, 1, 1, 0, 0) and (0, 0, 1, 1, 1) are represented in Figure 5. The induced colouring λ on a facet F takes its values in the vector space Figure 5: The induced colouring on the dart graphs associated with the edges (1, 1, 1, 0, 0) (left) and (0, 0, 1, 1, 1) (right).

V F = V / λ(F ) , which is a vector space over Z/2Z of dimension 4. A basis for this vector space is given by the colours associated with the three edges which have a common endpoint at the vertex S (resp. N ), together with the colour associated with any other edge which has N (resp. S) as a vertex, for instance (1, 1, 0, 1, 0), (1, 0, 1, 1, 0), (0, 1, 1, 1, 0) and (0, 1, 1, 0, 1).

Let us denote by H the hyperbolic 3-manifold defined by the induced colouring λ on the facets of P 4 . Once again, the group of admissible symmetries Adm λ (F ) coincides with the full symmetry group Symm(F ) ∼ = Z/2Z × S 3 , and the action is given by permutation of the coordinates.

We have a short exact sequence

0 → V ′ → Isom c (H) → Symm(F ) → 0, (3)
where Symm(F ) ∼ = Z/2Z × S 3 is the symmetry group of F .

Cusps

By a symmetry argument, it is clear that all the cusps of the manifold M are isometric to each other. The colours on each complete sub-graph K 4 ⊂ K 5 generate a 4-dimensional subspace V ′′ of V = (Z/2Z) 5 . Therefore, to each ideal vertex v of P 4 there correspond two cusps of the manifold M, for a total of ten cusps. Each cusp is tessellated by 2 4 = 16 copies of a unit Euclidean cube.

Recall that the colours on the vertex figure of P 4 correspond to the colours associated with a complete sub-graph K 4 ⊂ K 5 . A basis for V ′′ is given by the colours associated with any set of three edges spanning a complete sub-graph K 3 ⊂ K 4 , together with the colour associated with one of the remaining edges. The choice of such a basis allows us to identify V ′′ with (Z/2Z) 4 .

In Figure 6, we choose such a basis and represent the induced colouring directly on the cubical vertex figure of P 4 .

By applying Proposition 3.2, we conclude that the resulting cusps are all 3-dimensional tori. An explicit computation, analogous to that of Example 3.2 shows that the associated translation lattice is generated by translations along the vectors (2, 2, 0), (2, 0, 2) and (0, 2, 2). Figure 6: The induced colouring on the cubical vertex figure of P 4 , up to a suitable choice of a basis for the vector space V ′′ ∼ = (Z/2Z) 4 .

To conclude, we discuss the cusp shapes of the hyper-surfaces associated with the facets of P 4 , and how these intersect the cusps of M. Recall that each P 3 -facet F of the polytope P 4 has an induced colouring in the 4-dimensional vector space V ′ = V / λ(F ) , where λ(F ) is the colour assigned to the facet F . The vertex figures of the polyhedron P 3 are unit Euclidean squares. It turns out that the induced V ′ -colouring on the edges of each square generate a subspace V ′′′ of V ′ of dimension 3. basis for V ′′′ is given by the colours c 1 , c 2 , c 3 assigned to any three edges of K 4 ⊂ K 5 representing F , and the colour of the remaining edge is given by c 1 + c 2 + c 3 , as in Figure 7. Again, to each ideal vertex of a facet F ∼ = P 3 there correspond two cusps in the hyper-surface H F associated with F , for a total of six cusps. Each cusp is tessellated by 8 unit squares. The Euclidean cusp shape is that of a flat torus T , generated by translations along the vectors (2, 2) and (2, -2) represented in Figure 8. Notice that the cusp sections of the ambient four-manifold M can also be generated by translations along the vectors (2, 2, 0), (2, -2, 0) and (2, 0, 2). In other words, the cusp is obtained by identifying opposite faces of a slanted parallelepiped over a square base. Parallel copies of the base square correspond to cusp sections of the hyper-surface H F . Each cusp section of M is intersected by six cusp sections of the hyper-surfaces H F , F ∈ F(P 4 ), arranged in three parallel pairs, as shown in Figure 9.

It is useful to write down the equations for the linear subspaces which are naturally associated with the ideal vertices of P 4 . Given such a vertex v, the labels on the facets which are adjacent to v generate a co-dimension one linear subspace of V . Recall that each of these vertices is naturally labelled with an integer i ∈ {1, 2, 3, 4, 5}. The linear subspace associated to the vertex labelled i is defined by the equation

j =i x j = 0. (4)

Mutations

In this section, we shall show how to use mutations along hyper-surfaces in the manifold M in order to obtain new manifolds which are not isometric to the original one and have interesting cusp sections. Namely, we shall construct two singled cusped hyperbolic manifolds X and Y whose cusp sections are, respectively, an orientable S 1 -bundle over the Klein bottle and a three-torus. A mutation consists in cutting M open along a two-sided, totally geodesic hyper-surface H, choosing an isometry φ : H -→ H from one of the resulting totally geodesic boundary components to the other, and glueing back the two boundary components using the isometry φ. Clearly, this operation can be performed more than once as long as we pick a collection of mutually disjoint hyper-surfaces.

In the case of the manifold M constructed in Section 5, we can pick at most two disjoint totally geodesic hyper-surfaces corresponding to the facets of the polytope P 4 since, by Proposition 2.2, each such a facet F lifts to a unique hyper-surface in the orbifold cover M → P 4 , and there are at most two mutually disjoint facets in P 4 . Figure 9: A fundamental domain D for the cusps of the manifold M, together with its tessellation into copies of the unit cube C (shaded). The associated lattice is generated by translations along the vectors (2, 2, 0), (2, -2, 0) and (2, 0, 2). A cusp of a hyper-surface H F corresponds to the intersection of D with an affine hyperplane of the form z = c, with c integer.

It is useful to introduce labels on the cusps of M. In order to do so, we choose one of the copies, named P 0 , of the polytope P 4 that tessellates the manifold M. We label its ideal vertices, which naturally correspond to distinct sub-graphs K 4 ⊂ K 5 , with the numbers {1, 2, 3, 4, 5}.

The ideal vertices of P 0 determine the five cusps of M, which we label (1, +), (2, +), . . . , (5, +). Each other cusp of M is the image of exactly one of the cusps (1, +), (2, +), . . . , (5, +) under the action of the group V < Isom c (M), c.f. the short exact sequence (5.1). We label these remaining five cusps (1, -), (2, -), . . . , (5, -), respectively.

An orientable manifold with a single non-toric cusp

Let us pick two disjoint hyper-surfaces H 1 and H 2 in M, for instance those associated with the facets F 1 and F 2 having colours (1, 1, 1, 0, 0) and (0, 0, 1, 1, 1). To these two hyper-surfaces, there correspond a total of twelve square cusp sections (six to each). Each of the cusps of H i intersects exactly one cusp section of M. In particular, the cusps of H 1 (resp. H 2 ) lie in the cusps of M with labels (1±, 2±, 3±) (resp. (3±, 4±, 5±)). Each cusp section of the ambient manifold M intersects exactly one of the cusp sections of H 1 ∪ H 2 , with the exception of the cusps labelled (3, +) and (3, -), each of which intersects two cusp sections of H 1 ∪ H 2 .

We shall use mutations along the hyper-surfaces H i in order to build an orientable manifold X with one non-toric cusp and volume 2 • v m = 8π 2 /3. Let us cut the manifold M along H 1 and H 2 to produce a manifold M with four totally geodesic boundary components H ± i , i = 1, 2. The result of this operation is cutting open the cusp sections of M along the cusp sections of the hyper-surfaces H i that produces a total of twelve cusp sections. Each of these sections is a Euclidean manifold with totally geodesic boundary, homeomorphic to T × [0, 1], where T is the flat torus represented in Figure 8.

Recall that we have chosen a priori a root for the developing graph of the colouring, i.e. a copy P 0 of the polytope P 4 in the tessellation of M. We label by H + i the boundary components of M that intersects the polytope P 0 in the facet F i , i = 1, 2, respectively, and by H - i the other one. The cusps (1, ±), (2, ±), (4, ±), (5, ±), are cut open producing eight of the cusps of M. The cusps coming from (1, ±) and (2, ±) (resp. (4, ±) and (5, ±)) are bounded by the hyper-surface H + 1 (resp. H + 2 ) on one side and by H - 1 (resp. H - 2 ) on the other. These cusps are isometric to the product T × [0, 2]. The cusps of M labelled (3, +) and (3, -) are cut open to produce a total of four cusp sections, isometric to the product T × [0, 1] labelled (3a, ±) and (3b, ±). Here, the letter a corresponds to the cusp carried by H 1 , and the letter b corresponds to the cusp carried by H 2 . We call these the short cusps of M. Each of them is bounded by H + 1 (resp., H - 1 ) on one side, and by H + 2 (resp., H - 2 ) on the other. All other cusps are referred to as long ones.

The T -fibres of the short cusps are naturally tessellated by copies of the unit square. Notice that the colouring assigned to the sides of this square is obtained by considering the restriction of the colouring on the cubic vertex figure of P 4 labelled 3 to the four faces with colours different from (1, 1, 1, 0, 0) and (0, 0, 1, 1, 1), as in Figure 10. These colours generate a co-dimension 2 subspace W of V , defined by the equations:

x 1 + x 2 + x 4 + x 5 = 0, x 1 + x 2 + x 3 = 0. ( 5
)
Then the four affine subspaces of A (see Section 2, Remark 2.7) associated with the subspace W correspond to the cusps labelled (3a, ±) and (3b, ±) of M. They are given by the following equations:

• The cusp (3a, +) corresponds to

x 1 + x 2 + x 4 + x 5 = 0, x 1 + x 2 + x 3 = 0. (6)
• The cusp (3b, +) corresponds to

x 1 + x 2 + x 4 + x 5 = 0, x 1 + x 2 + x 3 = 1. (7)
• The cusp (3a, -) corresponds to

x 1 + x 2 + x 4 + x 5 = 1, x 1 + x 2 + x 3 = 0. (8)
• The cusp (3b, -) corresponds to

x 1 + x 2 + x 4 + x 5 = 1, x 1 + x 2 + x 3 = 1. (9)
Let us define a self-isometry of H 1 that we shall use in our construction. Consider the short exact sequence (5.2). Pick the isometry φ 1 = v • σ 1 , where σ 1 ∈ Symm (P 3 ) exchanges the hyperbolic vertices labelled 4 and 5 and induces the cycle (1, 2, 3) on the ideal vertices of Figure 3, and v = [(0, 1, 1, 0, 1)] ∈ V ′ . The translation by v corresponds to a reflection in the facet corresponding to the complete sub-graph K 3 ⊂ K 5 spanned by the vertices labelled 2, 3, and 4 in Figure 3.

Similarly, let us define a self-isometry of H 2 by φ 2 = σ 2 , where σ 2 ∈ Symm(P 3 ) fixes the hyperbolic vertices labelled 1 and 2, and induces the cycle [START_REF] Dufour | Notes on right-angled Coxeter polyhedra in hyperbolic spaces[END_REF][START_REF] Everitt | Right-angled Coxeter polytopes, hyperbolic six-manifolds, and a problem of Siegel[END_REF][START_REF] Fujiwara | Simplicial volume and fillings of hyperbolic manifolds[END_REF] on the ideal ones. Notice that both φ 1 and φ 2 are orientation-preserving isometries of H 1 and H 2 , respectively. Moreover, the maps φ 1 and φ 2 induce the following cycles on the cusps of M:

φ 1 : (1, +) → (2, +) → (3, +) → (1, -) → (2, -) → (3, -) → (1, +); ( 10
)
φ 2 : (3, ±) → (4, ±) → (5, ±) → (3±). ( 11
)
We glue isometrically in pairs the boundary components of M using the induced maps φ i :

H + i → H - i
to produce a manifold X .

Proposition 6.1. The manifold X is an orientable, finite-volume, single-cusped hyperbolic manifold. The cusp shape is a torus-bundle over S 1 , with the monodromy map given by a hyper-elliptic involution.

Proof. We prove that the cusp sections of M are glued together along their boundaries as in Figure 11 to produce a unique cycle of maximal length:

(3a, +) It is clear a priori that, no matter how many cusps the manifold X has, the corresponding cusp sections decompose into a union of 20 copies of the product T × [0, 1], glued along their boundaries in cycles to form a certain amount of closed Euclidean manifolds. Each such a cycle corresponds to a single cusp of X , and the shape of each cusp section is a torus bundle over S 1 . In order to compute the cycles, we choose a copy of T × [0, 1], and perform a sequence of reflections in parallel copies of the torus T of the form T × {0} and T × {1} at unit distance one from the other. All these reflections are induced by coloured isometries of the manifold M. In fact, there are two kinds of toric fibres in the cusp C: those which separate different cusps of the manifold M, and the central slices of the form T × {1} of the long cusps with labels (1±), (2, ±), (4±), (5, ±). Each of the latter cusps is isometric to T × [0, 2]. Each of the maps φ i : H i → H i admits a unique orientation-reversing extension φ i to an isometry in Isom c (M ) ∼ = V ⋊ S 5 as follows:

φ 1 -→ (1, -) φ 1 -→ (2, -) φ 1 -→ (3b, -) φ 2 -→ (4, -) φ 2 -→ (5, -) φ 2 -→ (3a, -) φ 1 -→ (1, +) φ 1 -→ . . . . . .
φ 1 :       a b c d e       →       c + 1 a b e d + 1       , φ 2 :       a b c d e       →       a b e + 1 c + 1 d + 1       (12)
Here we use the affine coordinates on the vertex set A of the developing graph Γ(M). These isometries of M induce reflections along the copies of T which separate different cusps of M.

The corresponding reflections along the central sections of the long cusps of M labelled i ∈ {1, 2, 4, 5}, can be expressed by the maps R i defined, in the affine coordinates on A, as follows:

R 1 :       a b c d e       →       a + 1 b c d + 1 e + 1       , R 2 :       a b c d e       →       a b + 1 c d + 1 e + 1       (13)
R 4 :       a b c d e       →       a + 1 b + 1 c d + 1 e       , R 5 :       a b c d e       →       a + 1 b + 1 c d e + 1       (14)
Notice that the map R i is simply a reflection in the hyper-surface which intersect the cusps of M labelled i ∈ {1, 2, 4, 5} and is parallel to H 1 ∪ H 2 . To each copy of T × [0, 2], there corresponds a co-dimension two affine subspace of A, determined by the eight copies of the polytope P 4 that tessellate the former. Since each reflection acts on the affine space A, the cycles induced on the 20 copies of T × [0, 1] naturally correspond to the cycles induced on the corresponding affine subspaces. Moreover, to each cycle C there corresponds naturally an element φ C in the coloured isometry group of the torus T , which can be described by the following short exact sequence:

0 → W → Isom c (T ) → D 4 → 0. ( 15
)
The isometry defined by φ C corresponds to the monodromy of the fibration 0 → T → C → S 1 → 0.

We define the isometries ψ 1 , ψ 2 : M → M as follows:

ψ 1 = φ 1 • R 2 • φ 1 • R 1 • φ 1 (16)
ψ 2 = φ 2 • R 5 • φ 2 • R 4 • φ 2 (17)
By substituting the respective affine expressions for each factor in the composition of maps given by ( 12) -( 13), we represent ψ 1 and ψ 2 as

ψ 1 :       a b c d e       →       a + 1 b + 1 c + 1 e + 1 d       , ψ 2 :       a b c d e       →       a b c + 1 d + 1 e + 1       (18)
Let us choose a cusp of M, e.g. labelled (3a, +). The map ψ 1 corresponds to a composition of five subsequent reflections along parallel copies of the torus T in the cusp C of X , where the first reflection is along the boundary component of (3a, +) corresponding to the hyper-surface H + 1 . Note that ψ 1 maps the cusp (3a, +) to (3b, -), and it does so by sending H + 1 to H - 1 and fixing H + 2 . Now we apply the map ψ 2 , i.e. we perform other five subsequent reflections, starting from the boundary component of (3b, -) corresponding to H + 2 . This maps (3b, -) to (3a, -), H - 1 to H + 1 and H - 2 to H + 2 . Again, we apply ψ 1 , mapping (3a, -) to (3b, +), H + 1 to H - 1 and fixing H - 2 . This time, we arrive at a cusp which is bounded by H - 2 on one side. Therefore we need to consider the pairing map φ -1 2 : H - 2 → H + 2 to start with. Thus, the last five reflection correspond to applying ψ -1 2 in order to return back to (3a, +) and close up the cycle. The monodromy map φ : T → T can therefore be expressed as the following composition of maps in Aut c (M ):

φ = ψ -1 2 • ψ 1 • ψ 2 • ψ 1 (19)
By substituting the corresponding expressions in the affine coordinates from (18), we get

As expected, this affine map fixes the affine space W associated with the slice T , defined by formula (5). In fact, in the affine coordinates, φ is a translation by vector (0, 0, 0, 1, 1). Notice that this vector is expressed as a sum of the colours assigned to the sides of the square in Figure 10 adjacent to its upper left vertex. Therefore, the resulting monodromy map is realised by a composition of two reflections along two orthogonal axes: it is a hyper-elliptic involution of the torus T . The lattice associated with the cusp C is generated by the translations along the vectors (2, 2, 0) and (2, -2, 0) together with the Euclidean isometry ψ : (x, y, z) → (-x, -y, z + 20).

An orientable manifold of small volume with a single toric cusp

Let us step back for a moment, and look once more at the symmetric manifold M defined in Section 5.1. There is an orientation-preserving involution i, exchanging the hyper-surfaces H 1 and H 2 associated with the facets having colours (1, 1, 1, 0, 0) and (0, 0, 1, 1, 1). In terms of the short exact sequence (5.1), it is given by i = σ, where σ = (1, 4)(2, 5) ∈ S 5 .

Again, we use mutations along the hyper-surfaces H 1 and H 2 in order to pair all the cusps of M. For the hyper-surface H 1 , we choose the isometry φ 1 : H + 1 → H - 1 introduced in Section 6.1. For the hypersurface H 2 , we notice that the involution i sends H + 1 (resp., H - 1 ) to H + 2 (resp., H - 2 ), and we conjugate the map φ 1 by the involution i. Finally, the mutations φ 1 and φ 2 produce a hyperbolic manifold which we denote by Y. Proposition 6.2. The manifold Y is an orientable finite-volume hyperbolic 4-manifold with a single cusp, whose cross-section is a three-dimensional torus.

Proof. In order to determine the homeomorphism type of the cusp of Y, we follow the same procedure as in the proof of Proposition 6.1. The affine expression for the orientation-reversing lift φ 1 of φ 1 is the same as before, and is given in [START_REF] Long | On the geometric boundaries of hyperbolic 4-manifolds[END_REF]. The affine expression for the orientation-reversing lift of the second mutation φ 2 is

φ 2 :       a b c d e       →       b a + 1 e c + 1 d       . (21)
One can check that these pairing maps arrange the cusps of M in a single cycle:

(3a, +)

φ 1 -→ (1, -) φ 1 -→ (2, -) φ 1
-→ (3a, -)

. . .

φ -1 1 --→ (1, +) φ -1 1 --→ (3b, -) φ -1 2 --→ (5, -) φ -1 2 --→ (4, -) φ -1 2 --→ (3a, +). ( 23
)
Let us define the maps ψ 1 and ψ 2 as the following compositions:

ψ 1 = φ 1 • R 2 • φ 1 • R 1 • φ 1 , (24)
ψ 2 = φ 2 • R 5 • φ 2 • R 4 • φ 2 , (25)
The affine expression for ψ 1 coincides with the one given by formula [START_REF] Serre | Cohomologie des groupes discrets[END_REF] in Section 6.1. The affine expression for ψ 2 is given below

ψ 2 :       a b c d e       →       b + 1 a c + 1 d + 1 e + 1       . (26)
Then the monodromy map φ of the cusp section of Y equals

φ = ψ -1 2 • ψ -1 1 • ψ 2 • ψ 1 . (27)
By substituting the affine expressions for ψ 1 and ψ 2 given in ( 18) and (26), we see that φ is actually the identity map of the torus T . As a consequence, the manifold Y has a single toric cusp C. The Euclidean cusp shape is determined by translations along the vectors (2, 2, 0), (2, -2, 0) and (0, 0, 20). Remark 6.3. The examples above answer affirmatively, in the orientable case, to Question 4.15 from [START_REF] Kolpakov | Symmetries of hyperbolic 4-manifolds[END_REF].

Figure 1 :

 1 Figure 1: The colouring λ of the cube C, faces are labelled with vectors from V

Figure 2 :

 2 Figure 2: Moving through copies of C by reflections in their faces, from C 0 to C N . The isometry ψ is a translation + half-turn rotation that brings us back from C N to C 0

Figure 4 :

 4 Figure 4: The dart graph associated with an edge e of K 5 .

Figure 7 :

 7 Figure 7: The induced colouring on the square vertex figure of P 3 , up to a suitable choice of a basis for the vector space V ′′′ ∼ = (Z/2Z) 3 .

Figure 8 :

 8 Figure 8: The cusp shape of the hyper-surface H F , together with its tessellation by unit squares (dashed).

Figure 10 :

 10 Figure 10: The restricition of the V -colouring on the vertex figure of the ideal vertex labelled 3 to a square section parallel to the hyper-surfaces H 1 and H 2

2 -

 2 -→ (3a, +). corresponding to the unique cusp C of the manifold X .

Figure 11 :

 11 Figure 11: The local cusp structure of the manifold X , as a result of cusp sections glueing in M. Coloured regions represent intersections with the hyper-surfaces H 1 and H 2 , and are each isometric to the torus T in Figure8. Notice that the cusps with labels (3a, ±) and (3b, ±) are shorter than others, and are bounded by different hyper-surfaces on the sides. This property is crucial, because it allows us to "mix" the cycles induced on the cusps by the maps φ 1 and φ 2 .

by a result of L. Potyagaȋlo and È. Vinberg[START_REF] Potyagaȋlo | On right-angled reflection groups in hyperbolic space[END_REF],

≤ n ≤ 4 2 by a result of G. Dufour[START_REF] Dufour | Notes on right-angled Coxeter polyhedra in hyperbolic spaces[END_REF] right-angled polytopes exist in H n for 2 ≤ n ≤ 12. Examples are known up to dimension 8[START_REF] Potyagaȋlo | On right-angled reflection groups in hyperbolic space[END_REF]

Notice that this labelling is coherent with the one expressed in (4): the cusps of M with labels (3.+) and (3, -) correspond to different affine subspaces over x 1 + x 2 + x 4 + x 5 = 0.

An isometry of M induced by summation with a vector v from V is a covering transformation, and restricts to an isometry acting on the lifts of each F i . Since H i is the unique lift of F i , it follows that the isometry induced by v restricts to an isometry of H i . Thus, the orientation-reversing isometry α of M defined by summation with the vector (0, 0, 1, 0, 0) naturally induces an isometry of M which maps the cusp (3a, ±) to (3b, ±). The isometry β : M → M defined by summation with the vector (1, 1, 0, 1, 0) induces an isometry of M, mapping the cusp (3a, +) to (3a, -) and (3b, +) to (3b, -).

In order to understand which boundary components of M bound each of the short cusps, we notice that the isometry α is an orientation reversing isometry of M which is orientation-preserving on the hyper-surfaces H i Indeed, α is induced by the vector (0, 0, 1, 0, 0) = (1, 1, 0, 0, 1)

where v i 's belong to the colours of the neighbours of F 1 (as shown in Figure 5 on the left), and w is the colour of F 1 itself. Thus, in the induced colouring of F 1 we have a sum of four colours from V w , and the isometry α of H 1 can be expressed as a composition of four reflections, which is orientation preserving on H 1 .

We can apply an analogous argument to show that α is orientation-preserving on H 2 , as well. Here, we use the fact that (0, 0, 1, 0, 0) = (1, 0, 0, 1, 1)

where v i 's belong to the colours of the neighbours of F 2 (as shown in Figure 5 on the right), and w is the colour of F 2 itself.

Therefore, the induced isometry of M permutes the boundary components H + i and H - i . The map β instead, is an orientation-reversing isometry of M which is orientation-reversing on H 1 and orientation-preserving on H 2 . Indeed (1, 1, 0, 1, 0) = (1, 0, 0, 1, 1)

w where all the vectors on the right-hand side belong to the colours of the neighbours of F 1 (see Figure 5, left), on one hand, and the v i 's belong to the colours of the neighbours of F 2 (see Figure 5, right), on the other, while w is the colour of F 2 itself.

The colour w is the colour of F 2 itself. Therefore the induced map on M fixes the boundary components H ± 1 and permutes H + 2 and H - 2 . As a consequence, the short cusps of M have the following boundary components

• The cusp (3a, +) is bounded by H + 1 on one side and by H + 2 on the other.

• The cusp (3b, +) is bounded by H - 1 on one side and by H - 2 on the other.

• The cusp (3a, -) is bounded by H + 1 on one side and by H - 2 on the other.

• The cusp (3b, -) is bounded by H - 1 on one side and by H + 2 on the other.


Alexander Kolpakov

Jun Murakami

The dual Jacobian of a generalised hyperbolic tetrahedron, and volumes of prisms

Keywords: hyperbolic polyhedron, Gram matrix, volume. 2 Preliminaries

come   L'archive ouverte pluridisciplinaire

Introduction

Let T be a generalised hyperbolic tetrahedron (in the sense of [START_REF] Murakami | A volume formula for hyperbolic tetrahedra in terms of edge lengths[END_REF][START_REF] Ushijima | A volume formula for generalised hyperbolic tetrahedra[END_REF]) depicted in Fig. 1. If the truncating planes associated with its ultra-ideal vertices do not intersect, we call such a tetrahedron mildly truncated, otherwise we call it intensely truncated. If only two of them intersect, we call such a tetrahedron prism truncated [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF]. Let us note that a prism truncated orthoscheme is, in fact, a Lambert cube [START_REF] Kellerhals | On the volume of hyperbolic polyhedra[END_REF]. The volumes of the tetrahedron and its truncations are of particular interest, since they are the simplest representatives of hyperbolic polyhedra. Over the last decade an extensive study produced a number of volume formulae suitable for analytic and numerical exploration [START_REF] Cho | On the volume formula for hyperbolic tetrahedra[END_REF][START_REF] Derevnin | A formula for the volume of a hyperbolic tetrahedron[END_REF][START_REF] Kellerhals | On the volume of hyperbolic polyhedra[END_REF][START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF][START_REF] Murakami | On the volume of hyperbolic and spherical tetrahedron[END_REF][START_REF] Ushijima | A volume formula for generalised hyperbolic tetrahedra[END_REF]. A similar study was done for the spherical tetrahedron [START_REF] Kolpakov | Volume formula for a Z 2 -symmetric spherical tetrahedron through its edge lengths[END_REF][START_REF] Murakami | The volume formulas for a spherical tetrahedron[END_REF], which can be viewed as a natural counterpart of the hyperbolic one. Many analytic properties of the volume formula for a hyperbolic tetrahedron came into view concerning the Volume Conjecture [START_REF] Kashaev | The hyperbolic volume of knots from the quantum dilogarithm[END_REF][START_REF] Murakami | The colored Jones polynomials and the simplicial volume of a knot[END_REF]. However, other geometric characteristics of a generalised hyperbolic tetrahedron T are also important and bring some useful information. In particular, Figure 1: Generalised hyperbolic tetrahedron Jac(T ), the Jacobian of T , which is the Jacobian matrix of the edge length with respect to the dihedral angles, is such. This matrix enjoys many symmetries [START_REF] Luo | 3-dimensional Schläfli formula and its generalization[END_REF] and can be computed out of the Gram matrix of T [START_REF] Guo | Calculus of generalized hyperbolic tetrahedra[END_REF]. In the present paper, we consider Jac ⋆ (T ), the dual Jacobian of a generalised hyperbolic tetrahedron T . By the dual Jacobian of T we mean the Jacobian matrix of the dihedral angles with respect to the edge length. Such an object behaves nicely when T undergoes both mild and intense truncation: the dual Jacobian of a prism truncated tetrahedron is an analytic continuation for that of a mildly doubly truncated one. Let us mention, that the respective volume formulae are also connected by an analytic continuation, in an analogous manner [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF][START_REF] Murakami | A volume formula for hyperbolic tetrahedra in terms of edge lengths[END_REF]. As an application of our technique, we give a volume formula for a hyperbolic n-gonal prism, c.f. [START_REF] Derevnin | The Coxeter prisms in H 3 " in Recent advances in group theory and low-dimensional topology[END_REF]. a ij and lengths ℓ ij , i, j ∈ {1, 2, 3, 4}, i < j. Depending on whether the vertex v k is proper (v k ∈ H 3 ), ideal (v k ∈ ∂H 3 ) or ultra-ideal (v k defines a polar hyperplane as described in [21, Section 3], c.f. Theorem 3.2.12), let us set the quantity ε k to be +1, 0 or -1, respectively. For each vertex v i of T let us consider the face F jkl opposite to it, where {i, j, k, l} = {1, 2, 3, 4}. The link L(v l ) of the vertex v l is either a spherical triangle (ε l = +1), a Euclidean triangle (ε l = 0) or a hyperbolic triangle (ε l = -1). Let us define the quantity b i jk as follows:

b i jk :=       
the plane angle of F jkl opposite to the edge e jk , if ε l = +1; zero, if ε l = 0; the length of the common perpendicular to the edges e jl and e kl of F jkl , if ε l = -1.

Here, we consider the face F jkl as a generalised hyperbolic triangle, for which the trigonometric laws hold as described in [START_REF] Cho | Trigonometry in extended hyperbolic space and extended de Sitter space[END_REF][START_REF] Guo | Rigidity of polyhedral surfaces -II[END_REF]. Let us also define a quantity µ i jk by means of the formula

µ i jk := b i jk 0 cos( √ ε l s)ds.
Let µ ′ i jk denote the derivative of µ i jk with respect to b i jk , which means that

µ ′ i jk = cos( √ ε l b i jk ).
Let σ kl denote the following quantity associated with an edge e kl , k, l ∈ {1, 2, 3, 4}, k < l,

σ kl := 1 2 e ℓ kl - 1 2 ε k ε l e -ℓ kl .
Let σ ′ kl denote the derivative of σ kl with respect to ℓ kl , so we have that

σ ′ kl = 1 2 e ℓ kl + 1 2 ε k ε l e -ℓ kl .
Let us define the momentum M i of the vertex v i opposite to the face F jkl , {i, j, k, l} = {1, 2, 3, 4} by the following equality (c.f. [6, VII.6]):

M i := µ i jk µ i jl σ kl .
The quantity above is well defined grace to the following theorem.

Theorem 1 (The Sine Law for faces) Let F jkl be the face of T opposite to the vertex v i , {i, j, k, l} = {1, 2, 3, 4}. Then F jkl is a generalised hyperbolic triangle and the following equalities hold:

µ i jk σ jk = µ i jl σ jl = µ i kl σ kl .
Let us also define the momentum M jkl of the face F jkl opposite to the vertex v i , {i, j, k, l} = {1, 2, 3, 4} by setting (c.f. [6, VII.6])

M jkl := µ j kl sin a ik sin a il . The quantity above is well defined, according to the following theorem.

Theorem 2 (The Sine Law for links) Let v i be the vertex of T opposite to the face F jkl , {i, j, k, l} = {1, 2, 3, 4}. Then L(v i ) is either a spherical, a Euclidean or a hyperbolic triangle and the following equalities hold:

sin a ij µ j kl = sin a ik µ k jl = sin a il µ l jk .
Both Theorem 1 and Theorem 2 are paraphrases of the spherical, Euclidean or hyperbolic sine laws (for a generalised hyperbolic triangle, see [START_REF] Guo | Rigidity of polyhedral surfaces -II[END_REF]). The following theorems are the cosine laws for a generalised hyperbolic triangle adopted to the notation of the present paper.

Theorem 3 (The first Cosine Law for faces) Let F jkl be the face of T opposite to the vertex v i , {i, j, k, l} = {1, 2, 3, 4}. Then F jkl is a generalised hyperbolic triangle and the following equality holds:

σ ′ kl = µ ′ i kl + µ ′ i jk µ ′ i jl µ i jk µ i jl .
Theorem 4 (The second Cosine Law for faces) Let F jkl be the face of T opposite to the vertex v i , {i, j, k, l} = {1, 2, 3, 4}. Then F jkl is a generalised hyperbolic triangle and the following equality holds:

µ ′ i jk = -ε l σ ′ jk + σ ′ jl σ ′ kl σ jl σ kl .
Theorem 5 (The Cosine Law for links) Let v i be the vertex of T opposite to the face F jkl , {i, j, k, l} = {1, 2, 3, 4}. Then L(v i ) is either a spherical, a Euclidean or a generalised hyperbolic triangle and the following equality holds: µ ′ j kl = cos a ij + cos a ik cos a il sin a ik sin a il .

Auxiliary lemmata

In the present section we shall consider various partial derivatives of certain geometric quantities associated with either the faces or the vertex links of a generalised hyperbolic tetrahedron T . These derivatives will be used later on in the computation of the entries of Jac ⋆ (T ).

Lemma 1 For {i, j, k, l} = {1, 2, 3, 4} we have

∂ℓ kl ∂b i kl = -ε j µ i kl M i , ∂ℓ kl ∂b i jk = -σ ′ jl µ i kl M i , ∂ℓ kl ∂b i jl = -σ ′ jk µ i kl M i .
Proof. According to the definition of σ kl , we have σ kl = 0 only in the following two cases: ε k = ε l = +1 and ℓ kl = 0, or ε k = ε l = -1 and ℓ kl = 0. In the former case, we have a degenerate tetrahedron with two proper vertices collapsing to one point. In the latter case the tetrahedron has two ultra-ideal vertices, whose polar planes are tangent at a point on the ideal boundary ∂H 3 . This is a limiting case, since in a generalised (mildly truncated) tetrahedron two polar planes never intersect or become tangent. Thus, we suppose that σ kl = 0. By taking derivatives on both sides of the first Cosine Law for faces, we get the following formulae:

σ kl ∂ℓ kl ∂b i kl = ∂σ ′ kl ∂b i kl = 1 µ i jk µ i jl ∂µ ′ i kl ∂b i kl = -ε j µ i kl µ i jk µ i jl , since ∂σ ′ kl ∂b i kl = σ kl ∂ℓ kl ∂b i kl and ∂µ ′ i kl ∂b i kl = -ε j µ i kl
by a direct computation. This implies the first identity of the lemma. Now we compute

σ kl ∂ℓ kl ∂b i jk = ∂σ ′ kl ∂b i jk = - ((µ ′ i jk ) 2 + ε l (µ i jk ) 2 )µ i jl µ ′ i jl + µ i jl µ ′ i jk µ ′ i kl (µ i jk µ i jl ) 2 = - µ ′ i jl + µ ′ i jk µ ′ i kl µ i jk µ i kl • µ i kl µ i jk µ i jl = -σ ′ jl µ i kl µ i jk µ i jl .
where we use the identity (µ ′ i jk ) 2 + ε l (µ i jk ) 2 = 1 and, as before, the fact that

∂µ ′i jk ∂b i jk = -ε l µ i jk .
Then the second identity follows. The third one is analogous to the second one under the permutation of the indices k and l.

Lemma 2 For {i, j, k, l} = {1, 2, 3, 4} we have

∂b j kl ∂a ij = ε i sin a ij M jkl ∂b j kl ∂a ik = ε i sin a ij M jkl µ ′ l jk , ∂b j kl ∂a il = ε i sin a ij M jkl µ ′ k jl .
Proof. By taking derivatives on both sides of the Cosine Law for links, we get the following formulae:

-ε i µ j kl ∂b j kl ∂a ij = ∂µ ′ j kl ∂a ij = - sin a ij sin a ik sin a il .
The first identity of the lemma follows.

Then we subsequently compute

-ε i µ j kl ∂b j kl ∂a ik = ∂µ ′ j kl ∂a ik = - cos a il + cos a ij cos a ik sin a ij sin a ik sin a ij sin a ik sin a il = µ ′ l jk sin a ij sin a ik sin a il .
The second identity follows. The third one is analogous under the permutation of the indices k and l. Now we shall prove several identities that relate the principal minors G ii , i ∈ {1, 2, 3, 4} of the Gram matrix G := G(T ) of the tetrahedron T with its face or vertex momenta.

Lemma 3 For {i, j, k, l} = {1, 2, 3, 4}, we have that det G ii = ε i M 2 jkl .
Proof. Let us perform the computation for G 11 and other cases will follow by analogy. We have that

det   1 -cos a14 -cos a13 -cos a14 1 -cos a12 -cos a13 -cos a12 1   = = det   1 -cos a14 -cos a13 0 sin 2 a14 -µ ′2 34 sin a13 sin a14 0 -µ ′2 34 sin a13 sin a14 sin 2 a13   = = (1 -(µ ′ 2 34 ) 2
) sin 2 a 13 sin 2 a 14 = ε 1 (µ 2 34 ) 2 sin 2 a 13 sin 2 a 14 = ε 1 M 2 234 . By permuting the set {i, j, k, l} = {1, 2, 3, 4}, one gets all other identities of the lemma.

Lemma 4 For {i, j, k, l} = {1, 2, 3, 4}, we have that

-det G = sin 2 a jk sin 2 a jl sin 2 a kl (M i ) 2 . Proof. Let us subsequently compute det G = det     1 -cos a 34 -cos a 24 -cos a 23 -cos a 34 1 -cos a 14 -cos a 13 -cos a 24 -cos a 14 1 -cos a 12 -cos a 23 -cos a 13 -cos a 12 1     = det     1 -cos a34 -cos a24 -cos a23 0 sin 2 a34 -µ ′1 23 sin a24 sin a34 -µ ′1 24 sin a23 sin a34 0 -µ ′1
23 sin a24 sin a34 sin 2 a24 -µ ′1 34 sin a23 sin a24 0 -µ ′1 24 sin a23 sin a34 -µ ′1 34 sin a23 sin a24

sin 2 a23     = sin 2 a 23 sin 2 a 24 sin 2 a 34 det   1 -µ ′ 1 23 -µ ′ 1 24 -µ ′ 1 23 1 -µ ′ 1 34 -µ ′ 1 24 -µ ′ 1 34 1   = sin 2 a 23 sin 2 a 24 sin 2 a 34 det   1 -µ ′ 1 23 -µ ′ 1 24 0 ε 4 (µ 1 23 ) 2 -σ ′ 34 µ 1 23 µ 1 24 0 -σ ′ 34 µ 1 23 µ 1 24 ε 3 (µ 1 24 ) 2   = sin 2 a 23 sin 2 a 24 sin 2 a 34 (ε 3 ε 4 -(σ ′ 34 ) 2 )(µ 1 23 µ 1 24 ) 2 = -sin 2 a 23 sin 2 a 24 sin 2 a 34 (M 1 ) 2 .
Here we used the Cosine Law for links in the second equality and the first Cosine Law for faces in the fourth equality. Also, we used the fact that for {i, j, k, l} = {1, 2, 3, 4} one has 1

-ε l (µ i jk ) 2 = (µ ′ i jk ) 2 (in the third equality) and σ 2 ij -(σ ′ ij ) 2 = ε i ε j (in the sixth equality).
All other identities of the lemma follow by permuting the set {i, j, k, l} = {1, 2, 3, 4}.

Dual Jacobian of a generalised hyperbolic tetrahedron

In this section we shall compute the entries of the dual Jacobian matrix Jac ⋆ (T ) of a generalised hyperbolic tetrahedron T .

Theorem 6 Let T be a generalised hyperbolic tetrahedron. Then

Jac ⋆ (T ) := ∂(ℓ
η := Π 4 i=1 ε i det G ii (-det G) 3 1/2 , D :=         σ 12 σ 13 σ 14 σ 23 σ 24 σ 34        
and

S :=         ω 12 ε 1 σ ′ 14 ε 1 σ ′ 13 ε 2 σ ′ 24 ε 2 σ ′ 23 1 ε 1 σ ′ 14 ω 13 ε 1 σ ′ 12 ε 3 σ ′ 34 1 ε 3 σ ′ 23 ε 1 σ ′ 13 ε 1 σ ′ 12 ω 14 1 ε 4 σ ′ 34 ε 4 σ ′ 24 ε 2 σ ′ 24 ε 3 σ ′ 34 1 ω 23 ε 2 σ ′ 12 ε 3 σ ′ 13 ε 2 σ ′ 23 1 ε 4 σ ′ 34 ε 2 σ ′ 12 ω 24 ε 4 σ ′ 14 1 ε 3 σ ′ 23 ε 4 σ ′ 24 ε 3 σ ′ 13 ε 4 σ ′ 14 ω 34        
, where

ω kl := σ ′ ik σ ′ jl + ε l σ ′ il σ ′ jl σ ′ kl + σ ′ il σ ′ jk + ε k σ ′ ik σ ′ jk σ ′ kl σ 2 kl .
Proof. We compute the respective derivatives, that constitute the entries of Jac ⋆ (T ). Suppose that ε i = 0, i ∈ {1, 2, 3, 4}, since the cases when ε j = 0 for some j ∈ {1, 2, 3, 4} can be dealt with in an analogous manner. Then for {i, j, k, l} = {1, 2, 3, 4}, one has

∂ℓ kl ∂a ij = ∂ℓ kl ∂b j kl ∂b j kl ∂a ij = -ε i µ j kl M j • ε i sin a ij M jkl (1) = - 1 M j sin a ij sin a ik sin a il = - 1 M j sin aij sin a ik sin a il 1 σij 1 σ kl σij σ kl (2) = - 1 M j sin aij sin a ik sin a il σij σ kl µ i jk µ i jl M i µ l ik µ l jk M l (3) = 8 - M ijk M ikl M ijl M jkl (-det G) 3 σ ij σ kl (4) = - Π 4 i=1 ε i det G ii (-det G) 3 σ ij σ kl = -η σ ij σ kl .
Here we used the definitions of vertex and face momenta, as well as Lemmata 1 and 4. Indeed, in [START_REF] Buser | Geometry and spectra of compact Riemann surfaces[END_REF] we have that M ijk = µ j kl sin a ik sin a il and in (2) we use the fact that

σ ij = µ l ik µ l jk M l , σ kl = µ i jk µ i jl M i . In (3) we use µ i jk = M ijk sin a jl sin a kl , µ l ik = M ikl sin a ij sin a jk , µ i jl = M ijl sin a jk sin a kl , µ l jk = M jkl sin a ij sin a ik ,
together with the identities of Lemma 4. In (4) we use Lemma 3. Analogous to the above, we compute for {i, j, k, l} = {1, 2, 3, 4},

∂ℓ kl ∂a ik = ∂ℓ kl ∂b i jk ∂b i jk ∂a ik = -ε k sin a ik M ijl µ i kl M i σ ′ jk = -ε k ε j det G jj M i sin a jk sin a jl sin a ik √ ε k det G kk 1 σ ik σ kl σ ik σ kl σ ′ jk = -ε k ε j det G jj M i sin a jk sin a jl sin a ik √ ε k det G kk µ i jk µ i jl M i µ j il µ j kl M j σ ik σ kl σ ′ jk = -ε k ε j det G jj √ ε k det G kk √ ε l det G ll √ ε k det G kk √ ε k det G kk √ ε i det G ii (-det G) 3 σ ik σ kl σ ′ jk = -ε k Π 4 i=1 ε i det G ii (-det G) 3 σ ik σ kl σ ′ jk = -ε k η σ ik σ kl σ ′ jk .
Finally, for {i, j, k, l} = {1, 2, 3, 4}, we compute the derivative

∂ℓ kl ∂a kl = ∂ℓ kl ∂b i jk ∂b i jk ∂a kl + ∂ℓ kl ∂b i jl ∂b i jl ∂a kl .
Since the two terms of the above sum are symmetric under the permutation of k and l, we may compute only the first one. The second one will be analogous. By Lemmata 1 and 2, we get

∂ℓ kl ∂b i jk ∂b i jk ∂a kl = -ε l µ i kl M i sin a il M ijk σ ′ jl µ ′ j ik (5) = -(σ ′ ik σ ′ jl + ε l σ ′ il σ ′ jl σ ′ kl ) µ i kl M i sin a il M ijk 1 σ il σ kl (6) = -(σ ′ ik σ ′ jl + ε l σ ′ il σ ′ jl σ ′ kl ) ε j det G jj M i sin a jk sin a jl sin a il √ ε l det G ll µ i jk µ i jl M i µ j ik µ j kl M j (7)
=

-(σ ′ ik σ ′ jl + ε l σ ′ il σ ′ jl σ ′ kl ) εj det Gjj (-det G) 3 √ ε l det G ll √ ε k det G kk √ ε l det G ll √ εi det Gii √ ε l det G ll = -(σ ′ ik σ ′ jl + ε l σ ′ il σ ′ jl σ ′ kl ) Π 4 i=1 ε i det G ii (-det G) 3 = -η (σ ′ ik σ ′ jl + ε l σ ′ il σ ′ jl σ ′ kl ).
Here, in (5) we used the second Cosine Law for faces and in (6) we used the equality M ikl = µ i kl sin a jk sin a jl together with Lemma 3. In [START_REF] Guo | Calculus of generalized hyperbolic tetrahedra[END_REF] we perform a computation analogous to (3). Thus, we obtain

∂ℓ kl ∂a kl = ∂ℓ kl ∂b i jk ∂b i jk ∂a kl + ∂ℓ kl ∂b i jl ∂b i jl ∂a kl = -η (σ ′ ik σ ′ jl + ε l σ ′ il σ ′ jl σ ′ kl ) -η (σ ′ il σ ′ jk + ε k σ ′ ik σ ′ jk σ ′ kl ) = -η ω kl σ 2 kl .
The proof is completed.

Dual Jacobian of a doubly truncated hyperbolic tetrahedron

Let us consider the case when T is a (mildly) doubly truncated tetrahedron depicted in Fig. 2 with dihedral angles θ i and edge lengths ℓ i , i ∈ {1, 2, 3, 4, 5, 6}. We suppose that the vertices cut off by the respective polar planes are v 1 and v 2 .

If T is mildly truncated then the formula from Theorem 6 applies. If T is a prism truncated tetrahedron, as in Fig. 3, with dihedral angles µ, θ i and edge lengths ℓ, ℓ i , i ∈ {1, 2, 3, 5, 6} then its Gram matrix is given by

G =     1 -cos θ 1 -cos θ 5 -cos θ 3 -cos θ 1 1 -cos θ 6 -cos θ 2 -cos θ 5 -cos θ 6 1 -cosh ℓ -cos θ 3 -cos θ 2 -cosh ℓ 1     ,
which is a slightly different notation compared to [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF][START_REF] Kolpakov | Erratum to: Volume of a doubly truncated hyperbolic tetrahedron[END_REF]. Each link L(v k ), k = 1, 2, is a hyperbolic quadrilateral with two right sameside angles, which can be seen as a hyperbolic triangle with a single truncated

i 1j , with i, j ∈ {2, 3, 4}, i = j, for b i 1j + √ -1 π 2 and each b i 2j , with i, j ∈ {1, 3, 4}, i = j, for b i 2j + √ -1 π 2 .
Thus, some of the vertex and face momenta become complex numbers. All the trigonometric rules of Section 2 still hold grace to [2, Section 4.3]. Computing the respective derivatives in a complete analogy to the proof of Theorem 6, we obtain the following statement.

Volume of a hyperbolic prism

Let α n denote the n-tuple (α 1 , . . . , α n ) with 0 < α k < π, k = 1, . . . , n.

Let β n and γ n be analogous n-tuples. Let Π n := Π n ( α n , β n , γ n ) be the hyperbolic n-sided prism depicted in Fig. 4, with the respective dihedral angles, as shown in the picture. Let S k , k = 1, . . . , n, be the supporting hyperplane for the k-th side face of the prism Π n (we start numbering the faces anti-clockwise from the side face adjacent to the angles α 1 , β 1 and γ 1 , γ 2 ), and let S 0 and S n+1 be those of the top and the bottom face, correspondingly. For each S k , k = 0, . . . , n + 1, let S + k be the respective half-space containing the unit outer normal to it. Let S - k = H 3 \ S + k . Then Π n = n+1 i=0 S - i . Let T := T (α, α ′ , β, β ′ , γ; ℓ) be the prism truncated tetrahedron depicted in Fig. 5. Here α, α ′ , β, β ′ and γ are the respective dihedral angles, ℓ is the length of the respective edge. The volume Vol T of the tetrahedron T is given by [12, Theorem 1] * . Let v(α, α ′ , β, β ′ , γ; ℓ) := Vol T (α, α ′ , β, β ′ , γ; ℓ) denote the respective volume function. Theorem 8 Let Π n = Π n ( α n , β n , γ n ) be a hyperbolic n-sided prism, as in Fig. 4. If p 0 p n+1 ⊂ Π n , then the volume of Π n is given by the formula

Vol Π n = n k=1 v(α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ℓ ⋆ ),
where ℓ ⋆ is the unique solution to the equation ∂Φ ∂ℓ (ℓ) = 0, with

Φ(ℓ) := πℓ + n k=1 v(α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ).
Let P k , k = 1, . . . , n, be the plane containing p 0 p n+1 and orthogonal to S k . First, we consider the case when p 0 p n+1 lies inside the prism Π n and the planes P k , k = 1, . . . , n, divide the prism Π n into n prism truncated tetrahedra, as shown in Fig. 6.

Then each P k meets the k-th side face of the prism Π n . Thus, the planes S 0 , S k , S k⊕1 and S n+1 together with P k and P k⊕1 become the supporting planes for the faces of a prism truncated tetrahedron, which we denote by T k . Each P k is orthogonal to S k , S 0 and S n+1 . The dihedral angles of T k inherited from the prism Π n are easily identifiable. Let µ k denote the dihedral angle along the edge p 0 p n+1 and let ℓ ⋆ be its length. Then we have

T k = T (α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ ⋆ ), k = 1, . . . , n. Clearly, Vol Π n = n k=1 Vol T k = n k=1 v(α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ ⋆ ).
Thus, we have to prove only the following statement.

Proposition 1 If the common perpendicular p 0 p n+1 is inside the prism Π n and each P k meets the respective side also inside Π n , k = 1, . . . , n, then the equation ∂Φ ∂ℓ = 0 has a unique solution ℓ = ℓ ⋆ , the length of p 0 p n+1 .

Proof. Let us consider the collection of prism truncated tetrahedra

T k = T (α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ), k = 1, . . . , n. Each pair {T k , T k⊕1 } of them
has an isometric face corresponding to the plane P k⊕1 . Indeed, each such face is completely determined by the plane angles (two right angles at the side of length ℓ, the angles α k and β k at the opposite side) and one side length. We obtain the prism Π n ( α n , β n , γ n ) by glueing the tetrahedra T k together along the faces P k , k = 1, . . . , n, in the respective order. Their edges of length ℓ match together, and one obtains a prism if the angle sum of the dihedral angles µ k , k = 1, . . . , n, along them equals 2π. We have that

∂Φ ∂ℓ = π + n k=1 ∂v ∂ℓ (α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ).
Since v is the volume function from [12, Theorem 1], then by applying the Schläfli formula [16, Equation 1] one obtains

∂Φ ∂ℓ = π - 1 2 n k=1 µ k .
Thus, whenever the tetrahedra T k constitute a prism, we have n k=1 µ k = 2π or, equivalently, ∂Φ ∂ℓ = 0. The length ℓ in this case is exactly the length of the common perpendicular p 0 p n+1 to the planes S 0 and S n+1 . The rest is to prove that ℓ = ℓ ⋆ is a unique solution. In order to do so, we shall show that ∂µ k ∂ℓ > 0, k = 1, . . . , n. By using Theorem 7 we get the following formulae for a prism truncated tetrahedron (as depicted in Fig. 3):

∂ℓ 2 ∂ℓ = -η sin µ k sinh ℓ 6 cosh ℓ 2 , ∂ℓ 3 ∂ℓ = -η sin µ k sinh ℓ 5 cosh ℓ 3 , ∂ℓ 5 ∂ℓ = -η sin µ k sinh ℓ 3 cosh ℓ 5 , ∂ℓ 6 ∂ℓ = -η sin µ k sinh ℓ 2 cosh ℓ 6 .
Note that the above derivatives are all negative. In our present notation it means that for each prism truncated tetrahedron T k , k = 1, . . . , n, the edges of the top and bottom faces inherited from the prism Π n diminish their length if we increase solely the parameter ℓ. Recall that T k = T (α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ), and let us denote

T ′ k := T (α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ ′ ) with ℓ ′ > ℓ.
Let ABCD be the top (equiv., bottom) face of T k , as shown in Fig. 7, and A ′ B ′ C ′ D ′ be the top (equiv., bottom) face of T ′ k . Since the dihedral angles accept for µ k and µ ′ k remain the same, the plane angles of ABCD at A, B, C and those of A ′ B ′ C ′ D ′ at A ′ , B ′ and C ′ are respectively equal. One sees easily that we can match then ABCD and A ′ B ′ C ′ D ′ such that B and B ′ coincide, the sides AB and A ′ B ′ , BC and B ′ C ′ overlap and the point D ′ lies inside the quadrilateral ABCD. Then the area of A ′ B ′ C ′ D ′ is less than that of ABCD. Equivalently, by the angle defect formula [1, Theorem 1.1.7], µ ′ k > µ k . Thus, ∂µ k ∂ℓ > 0, k = 1, . . . , n, and the proposition follows. However, there is a possibility that, although the common perpendicular p 0 p n+1 is entirely inside the prism Π n , one (or several) of the planes P k meets the respective S k partially outside of the face S k . First we consider the case when a single plane P k meets S k entirely outside, as depicted in Fig. 8. Like this, we obtain the figure shaded in grey, that consists of two triangular prisms sharing an edge. Second we consider the case when a single plane P k meets S k partially outside, as depicted in Fig. 9. Like this, we obtain a more complicated figure that consists of two tetrahedra sharing an edge (one of which has two truncated vertices). Thus the planes S 0 , P k , P k⊕1 , S k , S and S n+1 bound a "butterfly" prism. We put k = 1, for clarity. In the general case, k ≥ 2, one uses induction on the number of planes P k meeting S k outside of Π n . Here, some other cases of "butterfly" prisms are possible.

Proposition 2 If the common perpendicular p 0 p n+1 is completely inside the prism Π n , the plane P 1 meets the plane S 1 outside of Π n , and all other P k , k = 2, . . . , n, meet the respective side faces inside Π n , then the volume of the prism equals where ℓ ⋆ is the unique solution to the equation ∂Φ ∂ℓ (ℓ) = 0, with

Vol Π n = n k=1 v(α k , α k⊕1 , β k , β k⊕1 , γ k⊕1 ; ℓ ⋆ ),
Φ(ℓ) := πℓ + n k=1 v(α k , α k⊕1 , β k , β k⊕1 , γ k ; ℓ).
Proof. We start with the case of a "butterfly" prism depicted in Fig. 8. Let us observe that the "butterfly" prism T 1 overlaps with the subsequent prism truncated tetrahedron T 2 exactly on its part

T (o) 1 outside of Π n . The part of T 1 inside Π n , called T (i)
1 , contributes to the total volume of the prism. The volume of T (o) 1 is excessive in the respective volume formula and should be subtracted. In fact, we prove that

v(α 1 , α 2 , β 1 , β 2 , γ 2 ; ℓ ⋆ ) = V := Vol T (i) 1 -Vol T (o) 1 ,
which implies that the excess in volume brought by T 2 is eliminated by the term "-Vol T (o) 1 ". In order to do so, let us denote by θ the dihedral angle along the common edge of the triangular prisms T 1 . Let ℓ θ be the length of this edge. Let γ := γ 2 and let ℓ γ be the length of the vertical edge with dihedral angle γ. We know that ∂V ∂γ = -1 2 ℓ γ , by the structure of the volume formula for a prism truncated tetrahedron. Indeed, the function V does not correspond to the volume of a real prism truncated tetrahedron any more, however all the Figure 9: Another "butterfly" prism truncated tetrahedron T k metric relations defining the dihedral angles between the respective planes are preserved. Thus, after computing the derivative ∂V ∂ℓ analogous to [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF], we obtain the latter equality. Now we compute the respective derivatives for the parts of the "butterfly" prism T 1 . Observe that the parameter θ depends on γ, while we vary γ and keep all other dihedral angles fixed. Let us denote γ = π -γ for brevity. We have that ∂Vol T The above identities together with the fact that

∂ ∂γ = -∂ ∂γ imply that ∂ ∂γ 2 v(α 1 , α 2 , β 1 , β 2 , γ 2 ; ℓ ⋆ ) = ∂V ∂γ 2 .
By analogy, we can prove that

∂ ∂ξ v(α 1 , α 2 , β 1 , β 2 , γ 2 ; ℓ ⋆ ) = ∂V ∂ξ ,
for any ξ ∈ {α 1 , α n , β 1 , β n , µ 1 }. The volume formula for a prism truncated tetrahedron implies that by setting α 1 = α n = π/2 and β 1 = β n = π/2 we Figure 10: Parametrising the "butterfly" prism depicted in Fig. 8 get v(α 1 , α n , β 1 , β n , γ 2 ; ℓ ⋆ ) = 0. In the case of a "butterfly" prism T 1 , under the same assignment of dihedral angles, we have that the bases of the two triangular prisms become orthogonal to their lateral sides. Thus T (i) [START_REF] Buser | Geometry and spectra of compact Riemann surfaces[END_REF] and

T (o)
1 degenerate into Euclidean prisms, which means that their volumes tend to zero. Thus, we obtain the identity v(α 1 , α n , β 1 , β n , γ 2 ; ℓ ⋆ ) = V . The proof of the monotonicity for the function ∂Φ ∂ℓ (ℓ) is analogous to that in Proposition 1. However, since the part T contributes to the function v(α 1 , α n , β 1 , β n , γ 2 ; ℓ) with the negative sign, we have to replace the edge lengths ℓ 3 and ℓ 5 with -ℓ 3 and -ℓ 5 , respectively, as shown in Fig. 10. Then we recompute the respective derivatives of the lengths of the horizontal edges according to Theorem 7. We obtain that the lengths ℓ 2 and ℓ 6 diminish, as before, while the lengths ℓ 3 and ℓ 5 increase. This implies that the upper (resp., lower) triangular base of T (i)′ 1 can be placed entirely inside the upper (resp. lower) triangular base of T (i) [START_REF] Buser | Geometry and spectra of compact Riemann surfaces[END_REF] . By the area comparison argument, we have that µ ′ 1 > µ 1 . The inequality ∂µ 1 ∂ℓ > 0 follows. All other cases of "butterfly prisms" (e.g. that in Fig. 9) can be considered by analogy.

Remark. In the general case, when the common perpendicular p 0 p n+1 does not lie entirely inside the prism Π n , we expect that an analogue to Theorem 8 holds with an exception that the equation ∂Φ ∂ℓ (ℓ) = 0 may have several solutions. However, one of these solutions is geometric and yields the volume of Π n .

Modified volume formula

We modify the volume formula for a prism truncated tetrahedron from [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF], in order to reduce it to a simpler form. Indeed, the formula in [12, Theorem 1] uses analytic continuation and accounts for possible branching with respect to any variable a j = e ℓ , with some j ∈ {1, 2, . . . , 6}, and a k = e i θ k , for any k ∈ {1, 2, . . . , 6}\{j}. Usually, we put j = 4 for simplicity. However, the formula allows for intense truncation at any edge, since it is invariant under a permutation of the variables a l , l ∈ {1, . . . , 6}. In our case, given a prism Π n and its decomposition into prism truncated tetrahedra T i , i ∈ {1, . . . , n}, we know that only the common perpendicular p 0 p n+1 is produced by an intense truncation. Thus, we can always put j = 4 and, moreover, the variable a 4 will be the only one that might cause branching. In this case, we suggest a simplified version of the formula from [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF]Theorem 1]. This formula also has less numeric discrepancies and performs faster, if used for an actual computation. Let us put a k := e i θ k , k ∈ {1, 2, 3, 5, 6}, a 4 := e ℓ , and let U = U (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , z) denote

U := Li 2 (z) + Li 2 (a 1 a 2 a 4 a 5 z) + Li 2 (a 1 a 3 a 4 a 6 z) + Li 2 (a 2 a 3 a 5 a 6 z) -Li 2 (-a 1 a 2 a 3 z) -Li 2 (-a 1 a 5 a 6 z) -Li 2 (-a 2 a 4 a 6 z) -Li 2 (-a 3 a 4 a 5 z),
where Li 2 (•) is the dilogarithm function. Let z -and z + be two solutions to the equation e z ∂U ∂z = 1 in the variable z. According to [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF][START_REF] Murakami | On the volume of hyperbolic and spherical tetrahedron[END_REF], these are z -:= -q 1 -q 2 1 -4q 0 q 2 2q 2 and z + := -q 1 + q 2 1 -4q 0 q 2 2q 2 , where q 0 := 1+a 1 a 2 a 3 +a 1 a 5 a 6 +a 2 a 4 a 6 +a 3 a 4 a 5 +a 1 a 2 a 4 a 5 +a 1 a 3 a 4 a 6 +a 2 a 3 a 5 a 6 ,

q 1 := -a 1 a 2 a 3 a 4 a 5 a 6 a 1 - 1 a 1 a 4 - 1 a 4 + a 2 - 1 a 2 a 5 - 1 a 5 + a 3 - 1 a 3 a 6 - 1 a 6 ,
q 2 := a 1 a 2 a 3 a 4 a 5 a 6 (a 1 a 4 + a 2 a 5 + a 3 a 6 + a 1 a 2 a 6 + a 1 a 3 a 5 + a 2 a 3 a 4 + a 4 a 5 a 6 + a 1 a 2 a 3 a 4 a 5 a 6 ).

Given a function f (x, y, . . . , z), let f (x, y, . . . , z) | z=z - z=z + denote the difference f (x, y, . . . , z -) -f (x, y, . . . , z + ). Now we define the following function V = V (a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , z) by means of the equality The equality (1) holds because of the commutativity of the operations ℜ and ∂ ∂a 4 for the function -V + a 4 ∂V ∂a 4 + µ 2 log a 4 . The latter holds since a 4 = e ℓ is a real parameter. This implies that ∂f (T ) ∂µ = -ℓ 2 . By analogy to the proof of [12, Theorem 1], we can show that ∂f (T )

V := i 4 U (
∂θ k = -ℓ k
2 , and that if T degenerates into a right Euclidean prism, then f (T ) → 0. Thus, Vol T = f (T ) and the proposition follows. Also, we have the following way to determine the dihedral angle µ along the length ℓ edge coming from the intense truncation.

Numerical examples

Finally, we produce some numerical examples concerning an n-gonal (n ≥ 5) prism Π n with the following distribution of dihedral angles: the angles along the vertical edges are 2π 5 , the angles adjacent to the bottom face are π 3 , and those adjacent to the top face are π 2 . Indeed, such a prism Π n exists due to [8, Theorem 1.1]. Then we apply Theorem 8 for the cases n = 5, 6, 7, and perform all necessary numeric computations with Wolfram Mathematica R .

Figure 2 :

 2 Figure 2: Doubly truncated tetrahedron (mild truncation)

Figure 3 :

 3 Figure 3: Doubly truncated tetrahedron (prism truncation)

Figure 4 :

 4 Figure 4: The prism Π n ( α n , β n , γ n )

Figure 5 :

 5 Figure 5: The prism truncated tetrahedron T (α, α ′ , β, β ′ , γ; ℓ)

Figure 6 :

 6 Figure 6: The decomposition of Π n (top view, on the left) and the prism truncated tetrahedron T k (on the right)

Figure 7 :

 7 Figure 7: Prisms T k and T ′ k with top faces marked

Figure 8 :

 8 Figure 8: The decomposition of Π n (top view, on the left) and the "butterfly" prism truncated tetrahedron T k (on the right)

  formula [16, Equation 1].

  [START_REF] Kolpakov | Volume of a doubly truncated hyperbolic tetrahedron[END_REF] , ℓ 13 , ℓ 14 , ℓ 23 , ℓ 24 , ℓ 34 ) ∂(a 12 , a 13 , a 14 , a 23 , a 24 , a 34 ) = -η DS D,

	where

  2 , ℓ 3 , ℓ 5 , ℓ 6 ) ∂(ℓ, θ 1 , θ 2 , θ 3 , θ 5 , θ 6 ) = ∂(ℓ 12 , ℓ 34 , ℓ 13 , ℓ 23 , ℓ 24 , ℓ 14 ) ∂(a 12 , a 34 , a 13 , a 23 , a 24 , a 14 ) .

  a 1 , a 2 , a 3 , a 4 , a 5 , a 6 , z) -z The volume of a prism truncated tetrahedron T is given byVol T = ℜ -V + a 4 ∂V ∂a 4 log a 4 .

												∂U ∂z	log z	z=z -z=z +	.
	Proposition 3 Proof. Let us denote
				f (T ) = ℜ -V + a 4	∂V ∂a 4	log a 4 ,
	and compute the derivative
		∂ ∂ℓ	f (T ) +	µ ℓ 2	= a 4	∂ ∂a 4		f (T ) +	µ log a 4 2	=
		= a 4	∂ ∂a 4	ℜ -V + a 4	∂V ∂a 4	+	µ 2	log a 4	.
	∂V ∂a 4 + µ 2 has an a.e. vanishing derivative, c.f. the note in [12] after Theorem 1 saying that µ ≡ -2 ℜ(a 4 The function ℜ a 4 ∂V ∂a 4 ) mod π. Hence,
	∂ ∂ℓ	f (T ) +	µ ℓ 2	= a 4	∂ ∂a 4	ℜ -V + a 4	∂V ∂a 4	+	µ 2	log a 4	(1) =
				(1) = ℜ -a 4	∂V ∂a 4	+ a 4	∂V ∂a 4		+	µ 2	=	µ 2	.

  Proposition 4 The angle µ is given by Proof. We have µ ≡ -2ℜ a 4 ∂V ∂a 4 mod π, where 0 < µ < π and has an a.e. vanishing derivative. Then we compute ∂U (a 1 , . . . , a 6 , z ± (a 1 , . . . , a 6 )) ∂U (a 1 , . . . , a 6 , z ± ) ∂z log z ± = ∂U (a 1 , . . . , a 6 , z ± ) ∂a 4 + ∂z ± ∂a 4 ∂U (a 1 , . . . , a 6 , z ± ) ∂z

	µ ≡ -ℜ	i a 4 2	∂U (a 1 , . . . , a 6 , z) ∂a 4	z=z -z=z +	mod π.
		∂a 4		-	∂ ∂a 4	z ±
						-	∂z ± ∂a 4
	Therefore, we obtain
	µ ≡ -2ℜ a 4	∂V ∂a 4	mod π ≡ -ℜ	i a 4 2	∂U (a 1 , . . . , a 6 , z) ∂a 4	z=z -z=z +	mod π,
	where 0 < µ < π.

∂U (a 1 , . . . , a 6 , z ± ) ∂z = ∂U (a 1 , . . . , a 6 , z ± ) ∂a 4 ,

since, for some m ∈ Z,

z ± ∂U (a 1 , . . . , a 6 , z ± ) ∂z = 2π i m,

by the definition of z -and z + .

Acknowledgements. The authors gratefully acknowledge financial support provided by the Swiss National Science Foundation (SNSF project no. P300P2-151316) and the Japan Society for the Promotion of Science (Grant-in-Aid projects no. 25287014, no. 2561002 and Invitation Programs for Research project no. S-14021). The authors thank the anonymous referee for his/her careful reading of the manuscript and helpful comments.

* in Section 7 we give a simplified formula for the volume of T .

In order to avoid excessive branching in numerical computations, we use the modified parameters

in the formulae for U and V from Section 7.

It follows from the definition of q ′ i , i = 1, 2, 3, above that the quantity q ′2

1 -4q ′ 0 q ′ 2 is a real number, c.f. [20, Section 1.1, Lemma]. This fact prevents computational discrepancies and simplifies any further numerical analysis of the volume formula.

Each of the above prisms Π n can be subdivided into n isometric copies of a prism truncated tetrahedron T n . Indeed, T n is a prism truncated tetrahedron with angles θ 1 = 2π 5 , θ 2 = θ 3 = π 2 , θ 5 = θ 6 = π 3 , and µ = 2π n . By rotating it along the edge with dihedral angle µ, we compose the desired prism Π n . The graph of Vol T n , with n = 5, as a function of ℓ, is shown in Fig. 11 on the left. The graph of ∂Φ ∂ℓ (ℓ) for the same prism truncated tetrahedron T n is depicted in Fig. 11 on the right. We observe that the function ∂Φ ∂ℓ (ℓ) is indeed monotone and has a single zero ℓ ⋆ ≈ 0.50672....