Francesco Amoroso

Umberto Zannier

Some remarks concerning the rank of mapping tori and ascending HNN-extensions of abelian groups

Keywords: Mapping tori groups; diophantine approximation. Mathematics Subject Codes: 20F05, 11D04, 11J86

Let A be a matrix in GL d (Z) of infinite order. In a recent paper, G. Levitt and V. Metaftsis prove that for any sufficiently large integer n the matrix A n is not conjugated to a companion matrix. We first prove a local version of this theorem. Then, we give an effective statement, using diophantine methods. We also discuss several related problems.

Introduction

The present paper considers certain diophantine issues arising from the study of the rank of the so-called mapping tori groups. These groups have been studied in a number of papers. For more on this, we referred to the article [START_REF] Levitt | Rank of mapping tori and companion matrices[END_REF] of Levitt and Metaftsis who in fact pointed out to us the question which we now analyze.

Let A ∈ GL d (Z). The natural action v → Av induces a semidirect product

G = Z d A Z = Z d , t | tvt -1 = Av
where we identify the generator 1 of Z with A. Let OR(A) be the minimum number of vectors whose A-orbits generate Z d and let rank(G) be the minimum number of generators of G. In [START_REF] Levitt | Rank of mapping tori and companion matrices[END_REF] the Authors shows that Theorem 1.1 (Levitt-Metaftsis) rank(G) = 1 + OR(A) .

In particular, G can be generated by two elements if and only if A is conjugate in GL d (Z) to a companion matrix.

We recall that a companion matrix is one of the shape

       0 * 1 0 * . . . . . . * 1 0 * 1 *        .
See also the end of this section for equivalent properties.

Since the conjugacy problem is decidable in GL d (Z) (cf. [START_REF] Grunewald | Solution of the conjugacy problem in certain arithmetic groups[END_REF]), one can decide whether G has rank 2 or not.

Motivated by a topological result of J. Souto [START_REF] Souto | The rank of the fundamental group of certain hyperbolic 3-manifolds fibering over the circle[END_REF], they then prove: Theorem 1.2 (Levitt-Metaftsis) Let A ∈ GL d (Z) be of infinite order. Consider the family of finitely generated groups G n = Z d A n Z. Then there exists n 0 = n 0 (A) such that rank(G n ) > 2 for n ≥ n 0 . In other words, for n ≥ n 0 the matrix A n is not conjugate to a companion matrix in GL d (Z).

Their proof is based on the Skolem-Mahler-Lech Theorem on linear recurrence sequences. An alternative approach to this last result follows from a local argument in Z p which amount to reduction modulo p, using equations in S-units. This approach actually shows a bit more, upon excluding the matrices all of whose eigenvalues are roots of unity. Namely, under the assumption that some complex eigenvalue of A ∈ GL d (Z) has infinite order, we shall prove the finiteness of the set of n ∈ Z such that A n is conjugate to a companion matrix in GL d (F p ) for all primes p outside a prescribed (but arbitrary) finite set S. In section 2 we prove: Theorem 1.3 Let S be a finite set of prime numbers and let A ∈ M d (Z). Suppose that A has two nonzero eigenvalues whose ratio is not a root of unity. Then there are only finitely many integers n such that for all primes p ∈ S the reduction modulo p of the matrix A n is conjugate to a companion matrix in GL d (F p ).

Let us pause for some remarks on this statement.

Remark 1.4 i) Assume A ∈ GL d (Z).
If the ratio of any two eigenvalues of A is a root of unity, then in fact all the eigenvalues must be roots of unity, because their product is det A = ±1.

ii) We remark that the assumption on the eigenvalues is necessary here, and (if S = ∅) cannot be replaced with the weaker one that A ∈ GL d (Z) has not finite order. This is shown by examples like S = {l},

A = 1 1 0 1 .
We notice that A has infinite order, nevertheless each of the powers A l m , m ∈ N, is conjugate to a companion matrix over F p , for all p = l, and actually over the ring Z[1/l].

On the other hand, we note that in the special case S = ∅, the conclusion of the Theorem holds even if A ∈ GL d (Z) has all eigenvalues roots of unity, but it is not of not finite order, see Proposition 2.1 in section 2.

iii) We also remark that for d > 1 there are integral unimodular matrices which are conjugate to a companion matrix in GL 2 (F p ) for all p (and indeed in SL 2 (Z p ) for all p) but not conjugate to a companion matrix over Z. Thus Hasse principle does not hold in the present situation. We provide an explicit example at the end of section 2. This shows that, even in the case S = ∅, the finiteness predicted by the Theorem is a priori a stronger assertion than the finiteness of the set of integers n such that A n is conjugate to a companion matrix over Z.

We give other complements to the results of [START_REF] Levitt | Rank of mapping tori and companion matrices[END_REF]. In section 3 we prove the following effective version of Theorem 1.2: Theorem 1.5 Let A ∈ GL d (Z) be a matrix of infinite order. Let Z be the set of positive integers n such that A n is conjugate in GL d (Z) to a companion matrix. Then there exists an effective absolute constant c > 0 such that

max Z ≤ cd 6 (log d) 2 .
We could ask for a strong version of Theorem 1.2. Let A ∈ GL d (Z) of infinite order. It is true that there exists n 0 = n 0 (A) such that OR(G n ) is "large" for n ≥ n 0 ? Levitt and Metaftsis give a simple counterexample to this statement. It is enough to choose integers h, k > 1 such that k + h = d and matrices A 1 ∈ GL k (Z), A 2 ∈ GL h (Z) with A 1 of infinite order and with A 2 conjugate to a companion matrix of finite order m . The matrix

A = A 1 0 0 A 2 is then of infinite order, but OR(A n ) ≤ k + 1 for n ≡ 1 mod m.
In section 4 we investigate more closely this problem. We formulate a conjecture which predicts that for a "generic" matrix A we would have OR(A n ) = 2 for infinitely many integers n. We shall relate this conjecture to another one concerning algebraic numbers and we shall give some evidence for it.

In section 5, we partially answer another question posed in [START_REF] Levitt | Rank of mapping tori and companion matrices[END_REF]. Let A ∈ M d (Z) be nonsingular and consider the so-called ascending HNN-extension

Z d * A = Z d , t | tvt -1 = Av .
By this notation it is meant that Z d * A is generated by Z d and t subject to the relations written on the right. We remark that Z d * A is a semidirect product if and only if A ∈ GL d (Z).

Levitt and Metaftsis ask if one can generalize Theorem 1.1 to such groups. We generalize the method of [START_REF] Levitt | Rank of mapping tori and companion matrices[END_REF], characterizing the rank of Z d * A in terms of the A-orbits. Then we give a positive answer to their question for 2 × 2 matrices.

Notations Let O be a ring and let A ∈ M d (O) be a d × d matrix with coefficient in O. We denote by OR O (A) the minimum number of elements needed, such that their A-orbits generate O d .

Let k be a field and let A ∈ M d (k). Then it is well known that OR k (A) is equal to the number of invariant factors of A. In particular, OR k (A) = 1 if and only if A is conjugate in GL d (k) to a companion matrix. This is in turn equivalent to require that the eigenspace of A relative to any eigenvalue has dimension 1. It is also equivalent to the fact that the minimal polynomial of A has degree d.

In what follows we shall consider matrices A ∈ M d (Z). We simply write OR(A) for OR Z (A).

An alternative proof using equations in S-units

In this section we first state and prove a Proposition which shows special cases in which the conclusion of Theorem 1.3 holds, with different assumptions. Then we prove Theorem 1.3 and we give some further remarks. Proposition 2.1 Let A ∈ M d (Z) and assume that A has at least one eigenvalue counted with multiplicity at least 2. Then, for all integers n > 1 and for all primes p | n, the matrix A n is not conjugate to a companion matrix in GL d (F p ).

In particular, if all the eigenvalues of A are roots of unity and A has infinite order, then for n > 1 there exists a prime p such that the matrix A n is not conjugate to a companion matrix in GL d (F p ).

Proof. By assumption the characteristic polynomial f (t) ∈ Z[t] of A has a multiple factor, and we may write f (t) = g 2 (t)h(t) for suitable monic polynomials g, h ∈ Z[t], with deg g ≥ 1.

Take n > 1 and choose p to be any prime divisor of n. If A n was conjugate to a companion matrix over F p , its minimal polynomial over F p would have degree d, and the same would hold for A p (because

A n ∈ F p [A p ]). However g(A p )h(A p ) ≡ g p (A)h p (A) = (g p-2 (A)h p-1 (A))f (A) = 0 (mod p). But deg g(t)h(t) < d, a contradiction.
Concerning the last assertion of the Proposition, note that if all the eigenvalues of A are roots of unity and A has infinite order, then A has at least one eigenvalue counted with multiplicity at least 2; for otherwise A would be diagonalizable (over Q) and hence of finite order. It is now sufficient to choose p as any prime divisor of n and to apply the previous part.

Before proving Theorem 1.3 we briefly recall some facts concerning the places of a number field K. For more details, see [START_REF] Bombieri | Heights in Diophantine geometry[END_REF], chapter 1, section 2 . Any absolute value of K induces a topology on K. Two absolute values | • | 1 , | • | 2 on K are said to be equivalent if they induce the same topology. It may be proved that this happens if and only if there exists l > 0 such that |x| 1 = |x| l 2 for all x ∈ K. An equivalence class of absolute values is called a place. We say that a place is non-archimedean (or finite) if the corresponding absolute values satisfy the ultrametric inequality |x + y| ≤ max(|x|, |y|). In this case, the restriction of | | to Q is a p-adic absolute value for some rational prime p. Let v be a finite place and let | | be one of the corresponding absolute value. Then

R v = {x ∈ K, | |x| ≤ 1} is a local ring. Let K(v) be its residue field. Let α ∈ K. If α ∈ R v we call reduction of α (at v) the image α mod v of α in K(v)
; otherwise we say that the reduction of α is not defined.

Proof of Theorem 1.3. By assumption we may pick nonzero complex eigenvalues λ, ξ of A, such that their ratio is not a root of unity (so in particular λ = ξ). Consider the number field K = Q(λ, ξ) and let x λ , x ξ be respective eigenvectors of A in K d \ {0}.

We let Σ be the set of places of K obtained as the union of the following sets:

(a) the set of archimedean places and the set of places above primes in S;

(b) the set of those places at which either λ or ξ is not a unit;

(c) the set of those places at which the reductions of λ and ξ coincide;

(d) the set of those places at which the reductions of x λ or x ξ are not defined or are linearly dependent.

Then Σ is a finite set of places of K. Suppose now that n is an integer such that A n is conjugate to a companion matrix over F p , for every prime p ∈ S. Fix v ∈ Σ and denote with a tilde the reduction modulo v.

We contend that λn = ξn . In fact, assuming the contrary we deduce that Ãn has the linearly independent eigenvectors xλ , xξ relative to the same eigenvalue λn . But then, in view of the characterization recalled at the end of section 1, Ãn cannot be conjugate to a companion matrix over the residue field of v. On the other hand if this residue field has characteristic p, we have p ∈ S (because v ∈ Σ); so A n is conjugate to a companion matrix over F p ; this is a contradiction which proves the claim.

This conclusion may be reformulated as the assertion that η n := λ n -ξ n is a Σ-unit for such an integer n; on the other hand both λ n , ξ n are Σ-units, due to our choice of Σ (see property (b)). So each relevant integer n provides a solution (η n /λ n , ξ n /λ n ) to the Σ-unit equation X + Y = 1 (i.e. to be solved in in Σ-units X, Y of K). But this equation has only finitely many solutions, due to a well-known (rather deep) Theorem in the theory of diophantine equations (see [START_REF] Bombieri | Heights in Diophantine geometry[END_REF], chapter 5). Hence (ξ/λ) n can take only finitely many values. Since ξ/λ is neither zero nor a root of unity, this proves that n too takes values in a finite set. This proves the Theorem.

We remark that the above equation in Σ-units is of special type, and this allows an entirely elementary treatment in special cases (that is, without relying on the deep result alluded to in the proof).

We also remark that Baker's theory of linear forms in logarithms allows to find effectively the finite set of relevant integers n, provided A and S are given effectively. This shall be implicit in the effective treatment in the next section.

Before going ahead with such effective analysis, we give the example promised in Remark 1.4, iii). If we write z = (t, u) with coordinates t, u ∈ O, this may be rephrased saying that det(z,

Example 2.2 Let

Az) = 11t 2 + 7tu -3617u 2 ∈ O * (resp. 11t 2 + 7tu -3617u 2 = 1 in the case of SL 2 (O)).
Consider then the quadratic form

Q(T, U ) = 11T 2 + 7T U -3617U 2 of discriminant ∆ = 7 2 + 4 • 11 • 3617 = 397
• 401 (where 397, 401 are primes). We have 44Q(T, U ) = (22T + 7U ) 2 -∆U 2 . It is easily checked that Q(T, U ) represents 1 over any Z p . Indeed, by the usual Hensel's principle (see [START_REF] Serre | Cours d'arithmétique[END_REF], II.2.2) we have only to pay attention to the special cases p = 2, 11, 397, 401 and prove the solvability of the corresponding congruences (i.e. modulo 8, 11, 397, 401). For p = 2, use Q(1, 1) ≡ 1 (mod 8). In the remaining three cases, use respectively that Q(-3, -1) ≡ 1 (mod 11) and that 44 is a quadratic residue modulo 397 and modulo 401.

We conclude that A is conjugate to a companion matrix in SL 2 (Z p ) for all primes p.

√

∆ is an integer in Z bounded in absolute value by 2 11ω/∆, which is < 1. Therefore ρ = ρ , so ρ ∈ Z. But this is impossible since 11 is not a square, proving the claim.

3 An effective bound.

In this section we prove the effective version of Theorem 1.2 announced in the introduction.

Proof of Theorem 1.5. We first remark that, by Proposition 2.1 in section 2, we may assume that A has distinct eigenvalues λ 1 , . . . , λ d and that at least one of these eigenvalues is not a root of unity.

We now recall some arguments from [START_REF] Levitt | Rank of mapping tori and companion matrices[END_REF]. Define, as in the proof of Proposition 5.3 of op. cit., the linear recurrence sequences m → u

(k) m (0 ≤ k ≤ d -1) by A m = u (0) m A 0 + • • • + u (d-1) m A d-1 .
These sequences form a basis of the Q-vector space V of linear recurrence sequences associated to the characteristic polynomial of A. Let ∆ n (A) be the determinant of the matrix (u

(k) nm ) 0≤m,k≤d-1 . Let n ∈ N and v 0 ∈ Z d . Then det(v 0 , A n v 0 , . . . , A (d-1)n v 0 ) = ∆ n (A) det(v 0 , Av 0 , . . . , A (d-1) v 0 ) .
Let, as in the statement of the theorem, Z be the set of positive integers n such that A n is conjugate in GL d (Z) to a companion matrix. Thus n ∈ Z if and only if Z d is generated by the A n -orbit of a vector v 0 , which in turn implies that Z d is generated by the A-orbit of v 0 . Thus, if n ∈ Z then there exists v 0 ∈ Z d such that

| det(v 0 , A n v 0 , . . . , A (d-1)n v 0 )| = | det(v 0 , Av 0 , . . . , A (d-1) v 0 )| = 1 .
Hence |∆ n (A)| = 1. On the other hand, let us assume |∆ n (A)| = 1. We may assume that the set Z is not empty, otherwise the conclusion of the theorem is trivial. Hence there exists

v 0 ∈ Z d such that det(v 0 , Av 0 , . . . , A (d-1) v 0 ) = ±1. Thus det(v 0 , A n v 0 , . . . , A (d-1)n v 0 ) = ±1 which shows that n ∈ Z.
The previous discussion proves that n ∈ Z if and only if

|∆ n (A)| = 1. Let D n (A) be the Vandermonde determinant D n (A) = det(λ nm k+1 ) 0≤m,k≤d-1 . Since the recurrence sequences m → λ m k+1 (0 ≤ k ≤ d -1)
give rise to another basis of V , we see that

D n (A) = det(C)∆ n (A) for some C ∈ GL d (Q). Thus n ∈ Z if and only if |D n (A)| = |D 1 (A)| (remark that ∆ 1 (A) = 1).
Let n ∈ Z. We shall obtain a bound for n from a lower bound for |D n (A)| and from an upper bound for |D 1 (A)|. First we recall some definitions. Given two monic polynomials f , g ∈ Z[t] we denote by disc(f ) the discriminant of f and by res(f, g) the resultant of f and g. We let M (f ) ≥ 1 its Mahler's measure, i.e. the absolute value of the product of the roots of f lying outside the unit circle. We also denote by f [n] the polynomial whose roots are the n-th powers of the roots of f . Let f (t) be the characteristic polynomial of A. We factorize f over

Z as f = f 1 • • • f s . We let d j = deg(f j ). Then |D n (A)| 2 = s i=1 |disc(f [n] i )| × 1≤i,j≤s |res(f [n] i , f [n] j )| ≥ s i=1 |disc(f [n] i )| (3.1)
Observe that f

[n] j has only simple roots (since otherwise D n (A) = 0 = D 1 (A)) thus it is of degree d j . By the main result of [START_REF] Dubickas | On the discriminant of the power of an algebraic number[END_REF] (which rests on lower bounds in linear forms in two logarithms), there exists an absolute positive constant c 0 such that |disc(f

[n] j )| ≥ M (f j ) (dj -1)(n-c0d 6 j log dj log n) .
We remark that M (f j ) dj -1 ≥ M (f j ). This is clear if

d j ≥ 2. If d j = 1 then f j = x ± 1 (A ∈ GL(n, Z) implies f (0) = ±1) thus M (f j ) = 1, and again M (f j ) dj -1 = 1 = M (f j ).
Assume n > c 0 d 6 log d log n (otherwise our claim is satisfied). By the remark above

|disc(f [n] j )| ≥ M (f j ) n-c0d 6 log d log n . (3.2)
By (3.1), (3.2) and by the multiplicativity of Mahler's measure we obtain

|D n (A)| 2 ≥ M (f ) n-c0d 6 log d log n .
By Hadamard's inequality:

|D 1 (A)| ≤ M (f ) d-1 d ρ 2 1 +•••+ρ 2 k ≤ M (f ) d-1 d d 2 . Thus |D n (A)| = |D 1 (A)| implies (n -c 0 d 6 log d log n -2d + 2) log M (f ) ≤ 2d 2 log d . (3.3)
Since at least one of the eigenvalues of A is not a root of unity, f is not a product of cyclotomic polynomials. By a Theorem of Dobrowolski [START_REF] Dobrowolski | On a question of Lehmer and the number of irreducible factors of a polynomial[END_REF], there exists an absolute positive constant c 1 such that log M (f ) ≥ c 1 (log d) -3 .

(3.4)

Assume n > c 0 d 6 log d log n + 2d -2 (otherwise our claim is again satisfied). From (3.3) and (3.4) we have

n ≤ 2c -1 1 d 2 (log d) 4 + c 0 d 6 log d log n + 2d -2
from which we easily get log n ≤ c 2 log d and then

n ≤ cd 6 (log d) 2
for some absolute positive constant c. One could ask if for a "generic" matrix A ∈ M d (Z) it is true that OR(A n ) = 2 infinitely often. Let us discuss a bit this problem.

Most of our results are local. Thus it is convenient to introduce the following notations. Let A ∈ M d (Z). Given a prime number p we let OR p (A) = OR Fp (A mod p). We define OR loc (A) as the maximum of OR p (A) for p a prime. We remark that OR loc (A) ≤ OR(A).

It seems that it happens only in very special cases that OR loc (A n ) is maximal (= d) for all large n.

For instance, using Fermat's little Theorem as in the proof of Proposition 2.1, it is easy to prove the following. for q running over the prime powers ≤ d (we recall that log ψ(d) ∼ d by the Prime Number Theorem). Then OR(

A n ) = OR loc (A n ) = d for n > ψ(d).
Proof. Let f (t) = (t -a) d be the characteristic polynomial of A. Let n > ψ(d). Thus there exists a power q of a prime p, such that q | n and q > d. Then A q -a ≡ (A-a) q mod p. Since q > d we have A q ≡ a mod p. Since q | n we also have A n ≡ a n/q mod p. Thus the minimal polynomial of A n mod p is linear, which implies OR

p (A n ) = d. Thus OR loc (A n ) = d. Since OR loc (A n ) ≤ OR(A n ) ≤ d we also have OR(A n ) = d.
Similarly, the method of the proof of Theorem 1.3 shows:

Remark 4.2 Let A ∈ M d (Z).
Assume that A has two nonzero eigenvalues whose ratio is not a root of unity. Let r be the sum of the dimensions of their eigenspaces. Then

OR(A n ) ≥ OR loc (A n ) ≥ r for n ≥ ψ(d).
On the opposite side, we generalize a conjecture of Ailon and Rudnick [START_REF] Ailon | Torsion points on curves and common divisors of a k -1 and b k -1[END_REF] which would imply that for a "generic" A ∈ M d (Z) we had OR loc (A n ) = 2 infinitely often. 2) There are no finite places v of K such that three distinct α j have the same reduction mod v.

Then for infinitely many n there are no finite places v of K such that three distinct α n j have the same reduction mod v.

Remark 4.4 i) In the special case α = (1, a, b) with a, b ∈ Z multiplicatively independent and such that gcd(a -1, b -1) = 1, Conjecture 4.3 reduces to conjecture A of [START_REF] Ailon | Torsion points on curves and common divisors of a k -1 and b k -1[END_REF]. As for this special case, we have a numerical evidence for it. Moreover, its analogous in function fields should be a consequence of a result of Lang, as in op.cit.

ii) Note that condition 2) is obviously necessary, but not condition 1), as already remarked in [START_REF] Ailon | Torsion points on curves and common divisors of a k -1 and b k -1[END_REF]. Take for instance a be a non-zero integer, a = ±1 and let, as in op.cit, α = (1, a, -a) which trivially satisfies the conclusion of Conjecture 4.3 but not assertion 1). More generally, we are confronted with this curious phenomenon. All the examples of algebraic numbers for which we can prove that they satisfy the conclusion of Conjecture 4.3, do not satisfy assertion 1).

iii) Here is another example, which comes from a linear recurrence sequence suggested by C. Ballot. Let u n = -1+F n+1 , where F n is Fibonacci's sequence. Then u n satisfies the linear recurrence sequence associated to the polynomial f (t) = t 3 -2t 2 +1 = (t-1)(t 2 -t-1) with roots 1, α and β. Let n be an odd integer not divisible by 3. Then there are no finite places v of O Q(α) such that 1, α n and β n have the same reduction mod v. Indeed,

f [n] (t) := (t -1)(t -α n )(t -β n ) = (t -1)(t 2 -L n t + (-1) n ), where L n is Lucas' sequence L 0 = 2, L 1 = 1, L n = L n-1 + L n-2 .
Thus for n odd f [n] mod p has 1 as triple root if and only if p = 2 and L n is even. In turn, L n is even if and only if 3 | n.

iv) The above linear recurrence sequence provides an example of a companion matrix

A ∈ GL 3 (Z) such that OR(A n ) = 2 infinitely often. Let A =   0 0 -1 1 0 0 0 1 2  
be the companion matrix of f (t) = t 3 -2t 2 + 1. Let v = (0, 1, 0) and let u n = -1 + F n+1 as before. Then it easily see that

A n v = (-u n , -u n-1 , u n+1 ). Thus Zv ⊕ ZA n v is primitive if and only if gcd(u n , u n+1 ) = 1. This proves gcd(u n , u n+1 ) = 1 =⇒ OR(A n ) ≤ 2 .
An exercise on Fibonacci's number shows that, for n odd not divisible by 3, gcd(u n , u n+1 ) = 1. Thus for these integers, OR(A n ) ≤ 2. Since A has infinite order, by Theorem 1.2 we have OR(A n ) = 2 infinitely often.

The Conjecture 4.3 immediately implies:

Conjecture 4.5 Let A ∈ M d (Z) be nonsingular with characteristic polynomial f . Let us assume:

1) The roots of the f do not satisfy non-trivial multiplicative relations of zero degree.

2) For all prime number p the polynomial f mod p do not have irreducible factors of multiplicity ≥ 3.

Then OR loc (A n ) ≤ 2 for infinitely many n.

Proof of Conjecture 4.3 ⇒ Conjecture 4.5. Let f (t) = d j=1 (t -α j ) be the characteristic polynomial of A. Thus the characteristic polynomial f

[n] of A n is f [n] (t) = d j=1 (t -α n j ).
Let us assume Conjecture 4.3. Then for infinitely many n and for p prime the polynomial f [n] mod p has no irreducible factors of multiplicity ≥ 3. Since OR p (A n ) is equal to the number of invariant factors of A n , we deduce that OR p (A n ) ≤ 2. Thus OR loc (A n ) ≤ 2 infinitely often.

with g i ∈ G and b ≥ 0. vii) Assume now that G can be generated by k + 1 elements, say x 1 , . . . , x k+1 . By Remark 5.2 v) we can find n ≥ 0 and s i ∈ {±1} such that F n t (x i ) si ∈ G + for i = 1, . . . , k + 1. Since F t is an automorphism, F n t (x 1 ) s1 , . . . , F n t (x k+1 ) s k+1 generate again G . By Remark 5.2 vi) there exist g 0 , . . . , g k , ∈ G and b ≥ 0 such that G = g 1 , . . . , g k , g 0 t b . Since t ∈ G , by Remark 5.2 ii) we have 1 = χ(t) ∈ bZ which implies b = 1.

Proof of Theorem 5.1. We first show that rank(G ) ≤ OR (ϕ) + 1. Let k = OR (ϕ). Thus there exist g 1 , . . . , g k ∈ G and N ∈ N for which Im(ϕ N ) is contained in the subgroup generated by the ϕ-orbits of g 1 , . . . , g k . Let g ∈ G. Then t N gt -N = ϕ N (g) ∈ g 1 , . . . , g k , t . Thus g ∈ g 1 , . . . , g k , t and G = g 1 , . . . , g k , t .

We now show that OR (ϕ) + 1 ≤ rank(G ). Let rank(G ) = k + 1. By Remark 5.2 vii) there exist g 0 , . . . , g k ∈ G such that G = g 1 , . . . , g k , g 0 t . Let g ∈ G. Then there exist i 1 , . . . , i l ∈ {1, . . . , k}, λ 1 , . . . , λ l ∈ Z and µ 1 , . . . , µ l ∈ Z such that

g = (g 0 t) λ1 g µ1 i1 • • • (g 0 t) λ l g µ l i l = F ρ1 (g i1 ) µ1 • • • F ρ l (g i l )
µ l (g 0 t) ρ l where F = F g0t is the inner automorphism x → (g 0 t)x(g 0 t) -1 and where

ρ i = λ 1 + • • • + λ i (i = 1, . . . , l). Let N g ≥ 0 such that m i := ρ i + N g ≥ 0 for i = 1, . . . , l. Then, by Remark 5.2 i), ϕ Ng (g) = F Ng (g) = F m1 (g i1 ) µ1 • • • F m l (g i l ) µ l (g 0 t) ρ l = ϕ m1 (g i1 ) µ1 • • • ϕ m l (g i l ) µ l (g 0 t) ρ l .
By Remark 5.2 ii) we have 0 = χ(ϕ Ng (g)) = χ((g 0 t) ρ l ) = ρ l . Thus ϕ Ng (g) is in the subgroup generated by the ϕ-orbits of g 1 , . . . , g l . It is now enough to choose N = max g N g for g running over a finite system of generators of G.

From now on we fix G = Z d . We translate the assertion OR (ϕ) = 1 in term of local conditions. Given a prime p we denote by ϕ : F d p → F d p the reduction mod p of ϕ. For v ∈ Z d we let Λ v be the subgroup generated by v, ϕ(v), . . . , ϕ d-1 (v) and we denote by Λ v its reduction mod p.

Let K be a field and let ψ be an endomorphism of a d-dimensional K-vector space V . We recall that dim Im(ψ j ) = dim Im(ψ d ) for j ≥ d. (5.5)

Remark. Let det(Λ v ) := [Z 2 : det(Λ v )]. Condition (5.5) is obviously satisfied if p det(Λ v ). If p | det(Λ v ) then dim(Λ v ) ≤ d -1 and condtition (5.5) is equivalent to dim Im(ϕ d ) = dim ϕ d-1 (Λ v ).
Proof. Assume first OR (ϕ) = 1. Then by definition, there exist a vector v ∈ Z d and N ∈ N such that Im(ϕ N ) ⊆ Λ v . Then Im(ϕ N +d ) ⊆ ϕ d (Λ v ). Let p be a prime. By the remark preceding the theorem, dim Im(ϕ d ) = dim Im(ϕ N +d ) ≤ dim ϕ d (Λ v ) .

Assume now that there exists v ∈ Z d such that (5.5) holds for every prime p. Let p be a prime. Since ϕ d (Λ v ) ⊆ Im(ϕ d ) and since these F p -vector spaces have the same dimension,

Im(ϕ d ) = ϕ d (Λ v ) ⊆ Λ v .
Let b be the product of the primes dividing det(Λ v ). By Bezout's identity we easily see that Im(ϕ d ) ⊆ Λ v + bZ d .

By induction we deduce Im(ϕ dN ) ⊆ Λ v + b N Z d

for N ∈ N. We chose for N a natural number such that det(Λ v ) divides b N . Then

Im(ϕ dN ) ⊆ Λ v + b N Z d ⊆ Λ v + det(Λ v )Z d ⊆ Λ v .
We consider the following even special case: G = Z 2 , ϕ ∈ M 2 (Z) non-singular. In this case, the assertion (5.5) is equivalent to the following two statements:

1) p | det(Λ v ) ⇒ p | det(ϕ).
2) ϕ(v) ≡ 0 mod p ⇒ p | tr(ϕ). Indeed, assume that p satisfies (5.5). Let p | det(Λ v ). Then dim Im(ϕ 2 ) = dim ϕ 2 (Λ v ) < 2. Thus ϕ is not injective and p | det(ϕ). Moreover, if ϕ(v) ≡ 0 mod p then ϕ(Λ v ) = 0, thus dim Im(ϕ 2 ) = dim ϕ 2 (Λ v ) = 0 and ϕ is nilpotent mod p which in turn implies p | tr(ϕ).

Conversely, let p be a prime satisfying 1) and 2). We have already remarked that (5.5) is trivially satisfied if p det(Λ v ). Assume that p | det(Λ v ). By 1), p | det(ϕ). Thus dim Im(ϕ 2 ) ≤ 1. Assume first ϕ(v) ≡ 0 mod p. Since p | det(Λ v ), we must have ϕ(v) ≡ λv mod p with λ ≡ 0 mod p. Hence Λ v =< v > Fp and ϕ 2 (Λ v ) =< λ 2 v > Fp . Thus dim ϕ 2 (Λ v ) = 1 ≥ dim Im(ϕ 2 ). If ϕ(v) ≡ 0 mod p, then, by 2), ϕ is nilpotent mod p, and again dim ϕ 2 (Λ v ) = dim Im(ϕ 2 ) = 0.

We write v = (x, y). Then det(Λ v ) = Q(x, y) with Q(X, Y ) a quadratic form. Thus the existence of a vector v ∈ Z 2 which satisfies conditions 1) and 2) above translate into the following statements on Q. There exist x, y ∈ Z such that for p prime we have: This last requirement amounts to certain finitely many congruence conditions, depending explicitly only on ϕ.

p | Q(x,
It is now a well-known matter to decide about the existence of x, y ∈ Z satisfying these congruences and moreover such that Q(x, y) is composed only of primes dividing det(ϕ) (and one can also calculate such x, y if there exist any). Thus: Corollary 5.4 Let ϕ : Z 2 → Z 2 be an injective morphism. Then one can compute rank(Z 2 * ϕ ).

  On the other hand, suppose that Q(a, b) = ∈ {1, -1} = Z * for some integers a, b ∈ Z. Then we would have (22a + 7b) 2 -∆b 2 = 44 . Consider the unit ω := (399 + √ ∆)/2 of the ring of integers O ∆ of Q( √ ∆). Let also ξ = (22a + 7b + b √ ∆)/2 which is again in O ∆ since 22a + 7b and b have the same parity. Then ξξ = 11 , where a dash denotes conjugation in Q( √ ∆). We could then find an integer m so that 11/ω ≤ |ξω m | < √ 11ω. Putting ρ := ξω m we find |ρρ | = 11, whence the above inequalities yield 11/ω < |ρ | ≤ √ 11ω. In turn, this gives |ρ -ρ | < 2 √ 11ω. Finally, (ρ-ρ )

4 The rank> 2 problem

 2 Let d ≥ 3. As recalled in the introduction, Levitt and Metaftsis provide a family of examples of matrices A ∈ M d (Z) of infinite order and such that OR(A n ) = 2 for infinitely many n. One can choose A = a 0 0 B with a = ±1 and B ∈ M d-1 (Z) conjugate to a companion matrix of finite order. Note however that this forces A ∈ GL d (Z). See Remark 4.4, iv) for an example of a matrix A ∈ GL 3 (Z) such that OR(A n ) = 2 for infinitely many n.

Remark 4 . 1

 41 Let A ∈ M d (Z). Assume that A has only one eigenvalue. Let ψ(d) = q≤d q .

Conjecture 4 . 3 = 1

 431 Let K be a number field and let α 1 , . . . , α d ∈ O K be non-zero algebraic integers of K. Let us assume: 1) α 1 , . . . , α d do not satisfy non-trivial multiplicative relations of zero degree. (Namely, α m1 1 • • • α m d d for integers m 1 , . . . , m d not all zero but with m 1 + . . . + m d = 0.)

Theorem 5 . 3

 53 Let ϕ : Z d → Z d be an injective morphism. Then OR (ϕ) = 1 if and only if there exists a vector v ∈ Z d such that for all prime p dim Im(ϕ d ) = dim ϕ d (Λ v ) .

  y) =⇒ p | det(ϕ) ϕ(x, y) ≡ 0 mod p =⇒ p | tr(ϕ) .

  A be the 2 × 2 unimodular matrix Then A is conjugate in SL 2 (Z p ) to a companion matrix for all p, but it is not conjugate to a companion matrix in GL 2 (Z).Proof. It is of course possible that there are simpler numerical examples, especially if we replace SL 2 (Z p ) by GL 2 (Z p ); we have not pursued in the task of finding one.To say that A is conjugate to a companion matrix over a ring O means that there is a vector z ∈ O 2 such that z, Az form an O-basis of O 2 ; in turn, this just means that det(z, Az) belongs to the group O * of invertible elements in O. If actually det(z, Az) = 1, then A is conjugate to a companion matrix in SL 2 (O).

	A :=	196 3617 11 203	.

We remark that if A ∈ GL d (Z), condition 1) of Conjecture 4.5 forces the roots of f to be different from roots of unity.

Ascending HNN-extensions

Let G be a finitely generated abelian group and let ϕ be an injective endomorphism of G. We consider the ascending HNN-extension

Define OR (ϕ) as the least positive integer k such that there exist g 1 , . . . , g k ∈ G and N ∈ N for which Im(ϕ N ) is contained in the subgroup generated by the ϕ-orbits of g 1 , . . . , g k . Remark that OR (ϕ) = OR(ϕ) for ϕ ∈ Aut(G). The following theorem generalizes the first statement of [START_REF] Levitt | Rank of mapping tori and companion matrices[END_REF], Corollary 2.4. Before proving this result, we make some simple remarks on ascending HNN-extensions. Some of them will be needed in the proof of the theorem.

For y ∈ G we denote by F y the inner automorphism of G defined by F y (x) = yxy -1 . Thus ϕ is the restriction of F t to G.

Remark 5.2

i) Since G is abelian, for g, g 1 ∈ G we have F gt (g 1 ) = gϕ(g 1 )g -1 = ϕ(g 1 ).

ii) We note that every x ∈ G may be written as x = t -a gt b , with g ∈ G and a, b ≥ 0.

Even if this form is not unique, t -a gt b → b -a defines a morphism χ :

and write the euclidean division b 2 = qb 1 + r (q ≥ 0, 0 ≤ r < b 1 ). Since q ≥ 0, by Remark 5.2 iii) we have x q 1 = g 1 t qb1 for some g 1 ∈ G. Thus

. By Euclid's algorithm we deduce that there exist x ∈ G + and g ∈ G such that x 1 , x 2 = g, x . More generally, if x 1 , . . . , x k+1 ∈ G + , then x 1 , . . . , x k+1 = g 1 , . . . , g k , g 0 t b

We have not made any particular effort to generalize this statement to higher dimension.

We finally remark that the analogous of Theorem 1.2 still holds in ascending HNNextensions and it is indeed an easy corollary of Theorem 1.3.

Corollary 5.5 Let A ∈ M d (Z) be a nonsingular matrix of infinite order. Suppose that A has two eigenvalues whose ratio is not a root of unity. Consider the family of ascending HNN-extensions

Proof. Let S be the set of primes dividing the discriminant of A. By Theorem 1.3 there exists n 0 = n 0 (A) such that for all n ≥ n 0 the matrix A n is not conjugate in GL d (F p ) to a companion matrix for all p ∈ S. Let n ≥ n 0 and assume OR (A n ) = 1. Let p ∈ S. Then, by the choice of S, the matrix A is in GL d (F p ). Thus OR(A n ) = OR (A n ) = 1 and A n is conjugate to a companion matrix in GL d (F p ), contradiction. Thus for n ≥ n 0 we have OR (A n ) > 1 and, by Proposition 5.1, rank(G n ) > 2.


Chenghai Xu

Robert Chehab

Peter Sievers

Xavier Artru

Michel Chevallier

Olivier Dadoun

Vladimir M Strakhovenko

A POSITRON SOURCE USING AN AXIALLY ORIENTED CRYSTAL ASSOCIATED TO A GRANULAR AMORPHOUS CONVERTER

A non conventional positron source using the intense γ radiation from an axially oriented monocrystal which materialize into e+e-pairs in a granular amorphous converter is described. The enhancement of photon radiation by multi-GeV electrons crossing a tungsten crystal along its <111> axis is reported. The resulting enhancement of pair production in an amorphous converter placed 2 meters downstream, is also reported. Sweeping off the charged particles from the crystal by a bending magnet upstream of the converter allows a significant reduction of the deposited energy density. Substituting a granular target made of small spheres to the usual compact one, makes easier the energy dissipation. The deposited energy and corresponding heating are analyzed and solutions for cooling are proposed. The configurations studied here for this kind of positron source allow its consideration for unpolarized positrons for the ILC.

1-INTRODUCTION

The use of axially oriented crystals with GeV electron beams in channeling conditions provide powerful sources of (unpolarized) photons. These photons can, then, materialize into e-e+ pairs in the same crystal or in amorphous converters put downstream. Theoretical studies [START_REF] Baier | Radiation yield of high-energy electrons in thick crystals[END_REF] and simulations [START_REF] Artru | Positron source using channeling in a tungsten crystal[END_REF][START_REF] Artru | Axial channeling of relativistic electrons in crystals as a source for positron production[END_REF][START_REF] Artru | Advantages of axially aligned crystal in positron production at future linear collider[END_REF] were confirmed by experiments at CERN and KEK [START_REF] Chehab | Experimental study of a crystal positron source[END_REF][START_REF] Artru | experiment with a crystal-assisted positron source using 6 and 10 GeV electrons[END_REF][START_REF] Artru | Summary of experimental studies at CERN on a positron source using crystal effects[END_REF][START_REF] Suwada | Measurement of positron production efficiency from a tungsten monocrystal using 4 and 8 GeV electrons[END_REF][START_REF] Suwada | First application of a tungsten single crystal positron source at the KEKB factory[END_REF]. Separating the crystal-radiator from the amorphous-converter with a bending magnet in between to sweep off the charged particles, allowing only photons to impinge on the converter, presents very interesting features concerning the amount of energy deposited and for its density. [START_REF] Artru | Polarized and unpolarized positron sources for electron-positron colliders[END_REF]. For such device, called hybrid source, the amount of PEDD (Peak Energy Deposition Density) can be made lower than critical values (35 J/g, for W) avoiding the use ofa multitarget system; that was already proposed and adopted for CLIC for its baseline [START_REF] Artru | Positron sources using channeling: a promising device for linear colliders[END_REF]. However, the problem of very intense beams as for ILC may present a serious problem for the target survival. We have carried out a study on a particular kind of converter made of granular substance (W spheres) which present the advantages of easier thermal dissipation; it was already considered for targets submitted to high intensity proton beams and dedicated to neutrino factories (P.Sievers, [START_REF] Sievers | a He-gas cooled stationary target[END_REF]) . Some results concerning a hybrid target with a granular converter are presented here.

2-THE HYBRID SOURCE

A high energy electron beam (5 to 10 GeV) is directed along the crystal axis . Photons, electrons and positrons are generated. A sweeping magnet takes off the charged particles and only the photons impinge on the amorphous converter (figure 1). The distance may be about 2 meters. For the ILC, we consider an electron beam of 10 GeV with a transverse rms radius of 2.5 mm. A scheme of the hybrid source is presented on figure 1.

Fig. 1 The hybrid positron source

2-1 The crystal radiator

The idea to substitute an oriented crystal to the magnetic undulator in order to produce an intense source of photons was generated more than twenty years ago [START_REF] Chehab | Proc. IEEE Part.Accel.Conf[END_REF]. Two main arguments prevailed: the very strong fields available on the atomic rows of the crystal, equivalent to 10 3 Tesla and the short periods associated to the oscillating trajectories of the electrons along the crystal axes (~ 1 μm ). The latter characteristics allows to reach tens of Mev photon energy with a GeV electron beam incident on the crystal; that represents two orders of magnitude less energy than for an undulator having cm period.. The energy spectrum and the yield of the photons produced in the crystal depend, obviously, on the electron incident energy. For that, it is interesting to consider a figure of merit [START_REF] Baier | Radiation yield of high-energy electrons in thick crystals[END_REF] given by the ratio: R= I ch /I br [START_REF] Baier | Radiation yield of high-energy electrons in thick crystals[END_REF] whereI ch is the intensity of channeling radiation and I br , that of bremsstrahlung. This ratio depends on the electron energy and on the crystal. For instance, R=1 for the axis <111> at 0.7 GeV for W, 1.3 GeV for Si and 1.9 GeV for Ge. At much higher energies, R>> 1 and the crystal effects dominate on bremsstrahlung and R is growing very fast with the energy. The radiation yield depends also of the crystal thickness. Increasing the crystal thickness, one may meet saturation due to the multiple scattering and the decrease in energy of the radiating particles.

Another peculiarity of the channeling radiation is that the emitted photons are much softer in energy than those created by bremsstrahlung. Therefore, the channeling radiation may be more efficient than bremsstrahlung, even with R=1, providing a larger number of photons. For the application foreseen here, the ILC positron source, we have chosen an incident energy of 10 GeV and a W crystal, with <111> axial orientation and 1 mm thickness. To fulfill good channeling operation, the following conditions are required: -A mosaic spread lower than the channeling critical angle -An incident beam divergence, lower than the channeling critical angle. This critical angle is given by :

ψ c = [2Uo/E] 1/2 (2)
whereUo is the depth of the potential well created by the atomic rows and E the electron energy. For our conditions (10 GeV, <111> W axis) the critical angle is ~ 0.5 mrad.

2-2 The granular converter

The very intense incident electron beam considered in the linear collider projects (10 14 e-/second) requiresresistant targets. In the muon collider and neutrino factories projects, a similar situation is met with powerful proton beams impinging on targets to produce pions decaying intomuons and then into neutrinos. P.Sievers and P. Pugnat have proposed the use of granular targets made of a great quantity of small spheres [START_REF] Sievers | a He-gas cooled stationary target[END_REF]. The high ratio of surface/volume of the spheres (~ 3/r) makes easier the thermal dissipation. In the scheme considered for the converter of the hybrid source, the spheres are arranged in staggered rows. An example is shown on figure 2. The choice of the number of rows is related to two essential quantities:

-The required positron yield -The Peak Energy Deposition Density (PEDD) For that purpose, we have made a comparison of two choices of granular targets (r=1 mm and r= 0.5 mm) with a compact target 8 mm thick which was convenient for the required positron yield. This comparison is shown on table 1.

Table 1. Comparison of compact and granular targets

We can observe that the "3 layers" granular target has almost 6% less positron yield than the compact target but 20% lower PEDD. It looks more interesting from that point of view than the r=0.5 mm granular solution. These considerations may also be derived from figure 3,where the total yield and energy deposited normalized to the incident electron are reported in function of the target thickness expressed in mm or in number of layers.

Fig.3 Comparison of compact and granular targets

We can see that the deposited energy for the "3 layers" target is 446 MeV/ewhereas it is of 523 MeV/e-for the 8 mm compact target.

3-THE SIMULATIONS

The simulations have been operated with twoprogrammes:

-A programme(VMS) describing the crystal effects (channeling, coherent bremsstrahlung, pair creation,..) written by V.M.Strakhovenko [START_REF] Strakhovenko | [END_REF]. -The GEANT4 code which uses the ouputs of the first programme as event generator; results on positron production are provided by GEANT4. Crystal effects are also described by the programme FOT; the results agree with VMS..

3-1 The results on photons from the crystal

Fig. 4 The photon energy spectrum [E-=10 GeV; l= 1mm W crystal] The photon characteristics are obviously important for the positron generation. We represent on figure 4 the energy spectrum in the case of a 10 GeV incident electron beam on a 1 mm thick crystal target oriented on its <111> axis. On figure 5, we have represented the transverse characteristics of the photon beam impinging on the granular converter. These photons have been created by an electron beam having an rms radius of 2.5 mm and an energy of 10 GeV. The rms photon beam radius at the target is ~ 3.5 mm.

3-2 Simulation results on positrons 3-2-1 The positron beam

The positrons created in the granular target (3 layers) are captured by an Adiabatic Matching device (AMD) which exhibits a slowly decreasing axial magnetic field from the converter to the entrance of the first accelerating section. The field is tapering from 6 tesla to 0.5 tesla over 50 cm. We present on figure 6 the energy spectrum, transverse dimension and momentum and time distribution at the converter exit. The same kind of figure (figure 7) concernes the same parameters after the capture by the AMD and one meter acceleration . We can observe the phase space transformation due to the matching device converting the small dimensions and large transverse momentum at the converter into larger dimensions and smaller transverse momentum after the AMD and one meter acceleration.

3-2-2 Energy deposited and PEDD

The configuration chosen for the linear collider ILC is based on a pulse time structure modification before the target in order to decrease the power deposited per pulse; the nominal structure of ILC beam is then recuperated after the Damping Ring. For that purpose, we have chosen the configuration proposed by T.Omorifrom KEK [START_REF] Omori | POSIPOL[END_REF] and where the incident electron beam is made of minitrains of 100 bunches each with a periodicity of 300 Hz in a macropulse of 40 ms containing 13 minitrains. See figure 8. The first target submitted to the incident beam is the crystal; An estimation of the energy deposited and of the PEDD is necessary.

The 10 GeV electron beam with an rms beam radius of 2.5 mm deposit about 8 MeV/e-in the 1 mm thick W crystal. That represents a power of 160 watts. Cooling is foreseen. The energy deposition density has a peak (PEDD) of 0.35 GeV/cm3/e-.. That represents for the ILC beam a PEDD of 5.8 J/g for a minitrain of 2.10 12 impinginge-. The minitrains are separated by 3.3 ms, giving enough time to the relaxation of shock waves.

Another problem for the crystal concernsthe high rate of Coulomb scattering on the nuclei which could lead to dislodgements and affect the crystal structure. The fluence is 2.10 16 e-/mm 2 /hour. It needs about 100 hours to reach the fluenceobtained in the SLAC test in 1997 where no damage was observed on the crystal [START_REF] Chehab | Radiation damage study of a monocrystalline tungsten converter[END_REF]. Estimating the critical fluence one order of magnitude larger, leads to a working time of thousand hours without estimated damage. Annealing procedures helps to restore the crystal qualities. The crystal must be cooled. It can be mounted in a water cooled round frame (a) (radially constrained) or mounted on only one side (with verticalbut no horizontal expansion. The target unit (Crystal + Be-windows) could also be cooled by a He jet (b), the heat being evacuated through both sides faces of the crystal. This allows to devise a support for the crystal which provides the required stability in its orientation , but permits its unconstrained lateral thermal expansion, to reduce thermal stresses and fatigue.Fivecrystals can be put on a transversally movingframe: that will reduce the average power deposited whereas the PEDD per micropulse (0.6 µs) and per macropulse (40 ms) will not change. The fluence associatedwith radiation damages is decreased by a factor 5. This choice is compatible with the use of a goniometer in which a horizontal motion is foreseen withthe two rotations (H and V axes) .The device is shown on figure 9. The translation direction is perpendicular to the plane of the figure. The amorphous converter is the target where the energy deposition is very important due to the large number of secondaries. For the chosen system (300 Hz)we have determined the energydeposition density in the (x,z)plane; where z is the propagation axis. The spheres have 1 mm radius and correspond to a volumeof 4mm 3. The largest value (PEDD)is 1.8 GeV/cm 3 /e-; it is slightly lowerthan that of a compact target.The PEDD corresponding to a minitrain is about 30 J/g. The total energy deposited in thetarget is ~ 446 MeV/e-for the3 layers target. The deposited energy density is presented on figure 10. The vertical scale is in GeV/cm 3 /e-. Fig. 10 Energy deposition density in the 3 layers granular converter. Beam direction is indicated (z axis). Maximum density is at target exit.

4-HEATING AND COOLING FOR THE GRANULAR CONVERTERS

In granular targets, consisting of densely packed spheres with sizes smaller than the rmstransverse beam profile of 3.5 mm, only small and essentially constant temperature gradients will be created across each individual sphere. Henceforth,the spheres will be able to expand relatively freely and will be submitted to negligible thermal stresses. Thermally induced shocks can be neglected when: r/2<<t o .c; r: radius of the sphere of 1 mm, to: pulse duration of a micro pulse of 0.6 micro seconds; c: velocity of sound in Tungsten: c=4.10 3 m/s. For ILC this is satisfied in good approximation. Another important feature of the granular target is that the deposited heat can be evacuated rapidly and at the location of heat deposition by the cooling fluid passing between the spheres. Excluding a stationary target, two solutions have been considered:

-the rotating wheel -the pendulum An overview of the thermal problems associated to this kind of e+ source is found in [START_REF] Sievers | Positron Source using channeling with a granular converter[END_REF].

4-1 A granular target on a rotating wheel

A stationary target would receive a too large amount of deposited energy, leading to serious heating problems.To reduce the adiabatic temperature rises induced by one macro pulse, the beam energy can be diluted by sweeping, rotating the target. Since the deposited beam energy is concentrated withina diameter of about 1 cm ( see previous plot), with a linear velocity of the rim of the rotating wheel of about 3 m/s, the rim is displaced by 1 cm over 3.3 ms, so that the energy of each minitrain is separated from the adjacent, following minitrain. Only very little pile up occurs. The calculated maximum temperature rise in the target rim, resulting from the PEDD per minitrain, is not exceeding 222 K. Details on the wheel and on the target container (Be and Ti) are presented below.

Beryllium 1 mm thick is chosen for entrance and exit windows; Titanium 1 mm thick is chosen for upper and lower parts of the container. Energy deposited and PEDD have been evaluated for the windows; they are negligible for Ti, whereas for Be the PEDD represents 30.6J/g (downstream window) leading to sustainable temperature rise of about 15 °K/minitrain. The rotating wheel is represented on figure 11.To minimize the temperature rise in the Helium during its passage along the periphery, its path length should be minimized by increasing the number of spokes for the He inlets and outlets.

Fig. 11The rotating wheel

The minimum periphery of the rotating wheel would be 13 cm, i.e. a diameter of 4 cm, to distribute uniformly a macro pulse over its rim over 40 ms. This results in a rotation frequency of 23 Hz or 1400 rpm, which may be difficult to achieve with rotating seals for the He inlets and outlets.. Therefore a larger wheel with a diameter of 58 cm is considered. With a velocity of the rim of 3.25 m/s, allowing for comfortable separation between minitrains, this leads to a rotation at 1.786 Hz or 107 r.p.m. which will be much easier to engineer. 3.10 6 pulses will be accumulated per target over 100 days of operation. With this configuration and taking into account the repetition frequency of the macro pulses of 5 Hz, it results that, in the average, each sphere will be hit every 5 turns, i.e. every 2.8 s. The total, average power, to be removed from a wheel (as from a stationary target) is about 10 kW. With Helium, pressurized to 1 MPa and with an entrance velocity of 10 m/s, a He mass flow of 30 gr/s is required which would result in an average temperature rise of 84 °C at the exit of the He-flow. Temporary and locally, values of 250 °C in the He may, however, occur at the hottest sphere just after a minitrain. Clearly, the wheel must be made vacuum tight and resist to the internal He-pressure of 1 MPa. Moreover, with the above specified He-cooling, the time constant of the exponential decrease in temperature of an adiabatically heated sphere is about 100 ms. Thus, a sphere will be cooled to practically zero before being hit again after 2.8 s.

4-2

Eddy currents: interference of the magnetic fringe field of the flux concentrator with the adjacent rotating wheel has been studied for fast rotating wheels (I.Bailey). With the above velocity ~ 3m/s, this problem should be much reduced. Anyway, this must be confirmed by further studies.

4-3-The pendulum

To avoid rotating seals, wobbling or trolling targets have been devised at Durham ILC e+ Meeting in October 2009, where the displacement of the target structure from the outside into the vacuum is made via flexible, vacuum tightbellows. The injection ofthe cooling fluid can thus be ensured through a rigid, non rotating structure. Inthe following we consider a "Pendulum Target", where the required displacement and velocity is provided by the sinusoidal oscillation of the target (figure 12). The width of the target is 13 cm, providing space for 13 minitrains with a diameter of 1 cm each and being displaced, when the beam is hitting the target, at a velocity of about 3 m/s over +/-7.5 degrees. The total swing is +/-23 degrees, allowing for comfortable inversion of the direction of the movement during the "off beam" time of 160 ms. The same range in angle must be sustained by the bellows oscillating at 2.5 Hz. Since in this configuration the average time between hits of the same target spot is about 0.2 s as compared to the rotating wheel with 2.8 s, average temperatures higher by about 100 K will result. This might, if necessary, be compensated by an improved He cooling. Clearly, prototyping will be required, also to assess the life time of the spheres, the windows and the bellows, submitted to 4.10 7 . cycles over 100 days of continuous operation.

5-SUMMARY AND CONCLUSIONS

The hybrid target with a granular amorphous converter exhibits: • -a good quality positron beam (yield, emittance) generated by photons from channeling radiation -the PEDD for the granular converter is lower than for the compact one -an efficient thermal dissipation process due to the granular character.

In particular, the enlargement of the macro pulse to 40 ms duration in the above considered « 300 Hz-KEK-System » and leaving a gap of « no-beam » of 160 ms, sufficient for efficient damping, opens the possibility to reduce the thermal load from the macro pulse on the target by displacing it laterally through rotation or trolling it.

Lateral velocities of about 3 m/s for this displacement are sufficient to reduce the local energy deposition density by a factor of 13 (in one macropulse) . Using granular targets, consisting of an ensemble of Tungsten spheres of 2 mm diameter or below and cooled by a Helium gas stream passing between the spheres, tolerable temperatures for Tungsten, Helium and the target container are achieved. In addition to a rotating wheel, a pendulum target is considered, by which the use of rotating vacuum and He-seals is avoided. Both target types seem to represent viable solutions for the positron production in the considered ILC scheme.

Fig. 2 A

 2 Fig.2 A Granular target : 8 rows of r= 1 mm spheres.

Fig. 5

 5 Fig.5-a Fig.5-b Fig.5The transverse characteristics of the photon beam at the converter; (a) x-y distribution; (b) Marginal y distribution on x plane; beam dimensions are given in mm.

Fig. 6

 6 Fig.6 Transverse dimensions and momentum, energy spectrum and time distribution at the target exit

Fig. 8The scheme

  Fig. 8The scheme of the 300 Hz solution proposed by Omori et al.

Fig. 9

 9 Fig. 9 The crystal mounting with the two cooling options : water cooling and He jet cooling

Fig. 12The pendulum

  Fig. 12The pendulum for a granular target

Acknowledgements

The authors are indebted to T.Suwada, T.Kamitani, T.Omori, J.Urakawa (KEK), T.Takahashi (Hiroshima University), L.Rinolfi, A.Vivoli (CERN) for valuable discussions. Thanks also to B.Mazoyer (LAL) for his help.