A topological Tits alternative. (English) Zbl 1149.20039

Breuillard, Emmanuel; Gelander, Tsachik

Ann. Math. (2) 166, No. 2, 427-474 (2007).

A celebrated results by Tits says that any finitely generated linear group contains either a solvable subgroup of finite index or a non-commutative free subgroup. The authors notice that the Tits alternative can be reformulated in a slightly stronger manner, using the Zariski topology induced on \(\Gamma\leq\text{GL}(n,K)\) from the Zariski topology on \(\text{GL}(n,K)\): if \(\Gamma\) is finitely generated, then it contains either a Zariski open solvable subgroup or a Zariski dense free subgroup of finite rank.
The main purpose of this relevant paper is to prove an analog of the previous statement when the ground field, and hence any linear group over it, carries a more interesting topology than the Zariski topology, namely for local fields. The main result is the following: let \(k\) be a local field and \(\Gamma\) a subgroup of \(\text{GL}(n,k)\); then \(\Gamma\) contains either an open solvable subgroup or a dense free subgroup. This dichotomy strongly depends on the choice of the topology assigned to \(k\) and on the embebdding of \(\Gamma\) in \(\text{GL}(n,K)\) and cannot be generalized to arbitrary topology. Nevertheless a weaker dichotomy is proved if \(\Gamma\) is a finitely generated dense subgroup of a locally compact group: either \(\Gamma\) contains a free group on two generators which is nondiscrete in \(G\) or \(G\) contains an open amenable subgroup.
Several interesting applications are given to the theory of profinite groups and in dynamics. Answering a conjecture of Dixon, Pyber, Seress and Shalev, the authors show that if a finitely generated linear group \(\Gamma\) is not virtually solvable, then its profinite completion \(\widehat\Gamma\) contains a dense free subgroup of finite rank. Moreover, it is proved (as it was conjectured by Shalev) that if an analytic pro-\(p\) group \(G\) satisfies a coset identity with respect to some open subgroup, then \(G\) is solvable and satisfies an identity.
The applications in dynamics are related to questions concerning amenability. For example a generalized Connes-Sullivan conjecture is obtained: let \(\Gamma\) be a countable subgroup of a locally compact topological group \(G\); then the action of \(\Gamma\) on \(G\) by left multiplication is amenable if and only if \(\Gamma\) contains a relatively open subgroup which is amenable as an abstract group.
Other important applications concern the growth of leaves in Riemannian foliations. In particular it is proved: let \(F\) be a Riemannian foliation on a compact manifold \(M\); the leaves of \(F\) have polynomial growth if and only if the structural Lie algebra of \(F\) is nilpotent; otherwise, generic leaves have exponential growth.

Reviewer: Andrea Lucchini (Padova)

MSC:

20G25	Linear algebraic groups over local fields and their integers
20E07	Subgroup theorems; subgroup growth
22E50	Representations of Lie and linear algebraic groups over local fields
22F10	Measurable group actions
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
53C12	Foliations (differential geometric aspects)
20E18	Limits, profinite groups
43A07	Means on groups, semigroups, etc.; amenable groups
57R30	Foliations in differential topology; geometric theory

Keywords:

linear groups; algebraic groups; Tits alternative; local fields; profinite groups; amenable groups; Riemann foliations; open solvable subgroups; dense free subgroups