Let k be a local field, and Γ ≤ GLn(k) a linear group over k. We prove that either Γ contains a relatively open solvable subgroup, or it contains a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups.

Let k be a local field, and Γ ≤ GLn(k) a linear group over k. We prove that Γ contains either a relatively open solvable subgroup or a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups.

Surprising as it may be at first sight, there are a number of connections between the theories of finite semigroups and dynamical systems, both viewed in a broad sense. For instance in symbolic dynamics, ideas or analogies from the theory of finite automata find a natural setting for application in sofic systems [10, 24] and, even though not usually formulated in dynamical terms, the dynamical ...

Let G be a profinite group and let α be an automorphism of G. Then α is topologically intense if, for every closed subgroup H of G, there exists x ∈ G such that α(H) = xHx. Topologically intense automorphisms are automatically continuous, because they stabilize each open normal subgroup of the group on which they are defined. We denote by Intc(G) the group of topologically intense automorphisms...

Standard examples of inverse limits arise from sequences of groups, with maps between them: for instance, if we have the sequence Gn = Z/pZ for n ≥ 0, with the natural quotient maps πn+1 : Gn+1 → Gn, the inverse limit consists of tuples (g0, g1, . . . ) ∈ ∏ n≥0Gn such that πn+1(gn+1) = gn for all n ≥ 0. This is a description of the p-adic integers Zp. It is clear that more generally if the Gn a...