5 Cheeger Constant and Algebraic Entropy of Linear Groups

We prove a uniform version of the Tits alternative. As a consequence, we obtain uniform lower bounds for the Cheeger constant of Cayley grahs of finitely generated non virtually solvable linear groups in arbitrary characteristic. Also we show that the algebraic entropy of discrete subgroups of a given Lie group is uniformly bounded away from zero. In this note, we summarize some results whose full proofs will appear in [5]. 1. Free subgroups in linear groups Let K be an arbitrary field and Γ a subgroup of GLd(K) generated by a finite subset Σ. Assume Σ is symmetric (i.e. s ∈ Σ ⇒ s−1 ∈ Σ), contains the identity e, and let G = G(Γ,Σ) be the associated Cayley graph. The set Σn is the set of all products of at most n elements from Σ, i.e. the ball of radius n centered at the identity in G. We introduce the following definition: Definition 1.1. Two elements in a group are said to be independent if they generate a non-commutative free subgroup. The independence diameter of a Cayley graph G(Γ,Σ) is the quantity dΓ(Σ) = inf{n ∈ N, Σn contains two independent elements }. Similarly, we define the independence diameter of the group Γ to be dΓ = sup{dΓ(Σ), Σ finite symmetric generating set with e ∈ Σ}. The Tits alternative [11] asserts that either Γ is virtually solvable (i.e. contains a solvable subgroup of finite index) or Γ contains two independent elements, i.e. dΓ(Σ) < +∞ for every generating set Σ. The two events are mutually exclusive. Tits’ proof provides no estimate as to how close to the identity in G the independent elements may be found. We obtain: Theorem 1.1. (Uniform Tits alternative) Let Γ be a finitely generated subgroup of GLn(K). Assume that Γ is not virtually solvable. Then dΓ < +∞. This result improves a theorem of A. Eskin, S. Mozes and H. Oh who proved in [7] the analogous statement when free subgroup is replaced by free semigroup. Although only linear groups in characteristic zero where 1991 Mathematics Subject Classification. Primary 20Fxx; Secondary 53Cxx.