Counting overlattices in automorphism groups of trees

We give an upper bound for the number uΓ(n) of “overlattices” in the automorphism group of a tree, containing a fixed lattice Γ with index n. For an example of Γ in the automorphism group of a 2p-regular tree whose quotient is a loop, we obtain a lower bound of the asymptotic behavior as well. Nous donnons une borne supérieure pour le nombre uΓ(n) de “surréseaux” contenant un réseau fixé d’indice n dans le groupe d’automorphismes d’un arbre. Dans le cas d’un arbre 2prégulier T , et d’un réseau Γ tel que Γ\T soit une boucle, nous obtenons aussi une minoration du comportement asymptotique. Introduction. Given a connected semisimple Lie group G, the Kazhdan-Margulis lemma says that there exists a positive lower bound for the covolume of cocompact lattices in G. This is no longer true when G is the automorphism group of a locally finite tree. Bass and Kulkarni (for cocompact lattices, see [BK]) and Carbone and Rosenberg (for arbitrary lattices in uniform trees, see [CR]) even constructed examples of increasing sequences of lattices (Γi)i∈N in Aut(T ) whose covolumes tend to 0 as i tends to ∞. If Γ is a cocompact lattice in the group Aut(T ) of automorphisms of a locally finite tree T , there is only a finite number uΓ(n) of “overlattices” Γ ′ containing Γ with fixed index n ([B]). Thus a natural question, which was raised by Bass and Lubotzky (see [BL]), would be to find the asymptotic behavior of uΓ(n) as n tends to ∞. In [G], Goldschmidt proved that there are only 15 isomorphism classes of (3,3)-amalgams. Thus for lattices Γ in the automorphism group of a 3-regular tree T whose edge-indexed quotient is 3 3 , one has uΓ(n) = 0 for n big enough. Moreover it is conjectured by Goldschmidt and Sims that there is only a finite number of (isomorphism classes of) (p, q)-almalgams, for any prime numbers p and q. In this paper, we give two results: an upper bound of uΓ(n) for any cocompact lattice, and a surprisingly big exact asymptotic growth of uΓ(n) for a specific lattice Γ in the automorphism group of a 2p-regular tree. Theorem 0.1. Let Γ be a cocompact lattice in Aut(T ). Then there are some positive constants C0 and C1 depending on Γ, such that ∀n ≥ 1, uΓ(n) ≤ C0n C1 log 2(n). Theorem 0.2. Let p be a prime number and let T be a 2p-regular tree. Let Γ be a cocompact lattice in Aut(T ) such that the quotient graph of groups is a loop whose edge stabilizer is trivial 1 and whose vertex stabilizer is a finite group of order p.