Sieve methods in group theory I: Powers in linear groups

Abstract

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...

Sieve methods in group theory II: the mapping class group

“…The proof that the set of elements which satisfy condition (a) is exponentially small uses the 'large sieve method' (implicitly in [16] and explicitly in [9]). For our purpose, this method can be summarized in the following theorem which follows from Theorem B of [11] Theorem 2 Let be a finitely generated group and let P be a set of all but finitely many primes. Let (N p ) p∈P be a series of finite index normal subgroups of .…”

Super-approximation, I: $\mathfrak {p}$-adic semisimple case

“…In the past decade they have been found to be extremely useful in a wide range of pure math problems, e.g. affine sieve [BGS10,SGS13], sieve in groups [LM13], variation of Galois representations [EHK12], etc. (see [BO14] for a collection of surveys of related works and applications).…”

“…As it was observed in [BGS10], one of the important consequences of having spectral gap modulo primes (see Theorem 21) is the fact that the probability of hitting a proper subvariety decays exponentially (see Proposition 30). Subsets of a finitely generated group with this property are called exponentially small by Lubotzky-Meiri [LM13]. Since the set of non-regular elements of a semisimple group is a proper subvariety, it was observed in [LM13] that the set of non-regular elements is an exponentially small set.…”

“…Subsets of a finitely generated group with this property are called exponentially small by Lubotzky-Meiri [LM13]. Since the set of non-regular elements of a semisimple group is a proper subvariety, it was observed in [LM13] that the set of non-regular elements is an exponentially small set.…”

SIEVE METHODS IN GROUP THEORY III: Aut(Fn)

“…‡ Current address: Institute for advanced Study, Princeton, NJ 08540, USA. One of the first applications of the large sieve method in group theory was a result of Rivin [12] (see also [5,6] for a comprehensive development of this method). He proved that the set of non-pseudo-Anosov elements in the Mapping Class Group, (MCG), is exponentially small (see also [9]).…”