# A topological Tits alternative

### Abstract

Let $k$ be a local field, and $\Gamma \leq \mathrm{GL}_{n}(k)$ a linear group over $k$. We prove that $\Gamma$ contains either a relatively open solvable subgroup or a relatively dense free subgroup. This result has applications in dynamics, Riemannian foliations and profinite groups.

## Authors

Emmanuel Breuillard

Le centre de mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France

Tsachik Gelander

Department of Mathematics, The Hebrew University, 91904 Jerusalem, Israel