We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n − 1, 1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n F n−1 are thin.
The sieve method is a classic one in number theory (see, for example, [FI]).Recently it found some applications in a non-commutative setting. On the one hand, Bourgain-Gamburd-Sarnak [BGS1] applied it in studying almost-prime vectors in orbits of non-commutative groups acting on Z n . On the other hand, Rivin [Ri] and Kowalski [Ko] used it to study generic properties of elements in the mapping class group and arithmetic groups. Our formulation of the sieve method generalizes and simplifies the second one and usually falls under the name 'Large Sieve'. The goal of this introduction is to state the general large sieve setting with respect to group theory and to serve as a guideline for the proof of Theorem A below. We start by describing the algebraic problem and its background. After that, we explain how random walks and sieve methods are used in its solution.A virtually nilpotent group is a group which contains a nilpotent subgroup of finite index. Mal cev proved: Theorem 1.1 (Mal cev [Mal]). Let Γ be a finitely generated virtually nilpotent group. Then, for every m ≥ 1 the set of m-powers Γ m := {g m | g ∈ Γ} contains a finite index subgroup of Γ.The converse is not true, not even for finitely generated groups: For every prime p, Golod and Shafarevich gave an example of a finitely generated residually finite infinite group Γ such that the order of every element of Γ is a power of p (see [Go] and [GS]). In particular, if m is coprime to p, then any element of Γ is an m-power.Yet, in [HKLS], two kinds of partial converse results were proved: Theorem 1.2 (Hrushovski-Kropholler-Lubotzky-Shalev [HKLS]). Let Γ be a virtually solvable group. If n ≥ 2 and n m=2 Γ m contains a finite index subgroup of Γ, then Γ is virtually nilpotent. On the other hand, there exists a solvable group Γ which is not virtually nilpotent and still there exists m ≥ 2 such that Γ m contains a coset of a finite index subgroup. Theorem 1.3 (Hrushovski-Kropholler-Lubotzky-Shalev [HKLS]). Let Γ be a finitely generated linear group. If n ≥ 2 and finitely many tessellates of n m=2 Γ m cover Γ, then Γ is virtually solvable. The formulation of the second theorem of [HKLS] suggests a stronger result; i.e., does the fact that finitely many tessellates of the set of all proper powers ∞ m=2 Γ m License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1120 ALEXANDER LUBOTZKY AND CHEN MEIRI cover Γ imply that Γ is virtually solvable? But the methods of [HKLS] are not suitable for handling all proper powers together. The reason is that the proof uses only local data of Γ, i.e., the images of n m=2 Γ m in finite quotients of Γ, and it is clear that the image of ∞ m=2 Γ m in such a quotient is the full quotient group (take m to be coprime to the size of the finite quotient). Thus, to extend the theorem to the case of all proper powers one need to combine the local data with some global data on the group and its elements. A way to do this is to use random walks, as we shall explain below.A finite subset Σ o...
If Γ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example SL n (Z), n ≥ 3) and Λ is a finitely generated group that is elementarily equivalent to Γ, then Λ is isomorphic to Γ. PreliminariesThe following is a theorem of Malcev. For completeness, we include a proof.Proposition 2.1 ([Mal]). If Λ is a group that is elementarily equivalent to a linear group, then Λ is linear. If, in addition, Λ is finitely generated, then Λ is residually finite.Proof. Let Φ ⊂ GL n (k) be a linear group which is elementary equivalent to Λ. Enumerate the elements of Λ as Λ = {λ α } α∈A , and enumerate all relations that hold between the λ α s by {r β (λ α )} β∈B . Let L be the first-order language of rings together with constants c α i,j , for α ∈ A and 1 ≤ i, j ≤ n. Consider the theory T consisting of the following statements:
We prove that the set of non-pseudo-Anosov elements in the Torelli group is exponentially small. This answers a question of Kowalski (2008).
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