In his paper on free subgroups of linear groups, Tits proved his famous alternative: a linear group is either virtually solvable or contains a free subgroup. Since then, Tits’ work has been generalized and applied in many different ways.

One remaining open question in this field was the one asked by de la Harpe: let $G$ be a semisimple Lie group without compact factors and with trivial center and let $\Gamma$ be a Zariski-dense subgroup of $G$. Given a prescribed finite subset $F$ of $G$, is it always possible to find an element $\gamma \in \Gamma$ such that for any $h \in F$, the subgroup generated by $h$ and $\gamma$ is freely generated (in that case, we say $h$ and $\gamma$ are ping-pong partners).

In the talk, we will discuss a variant of the question of de la Harpe, where $F$ is a finite set of finite subgroups $H_i$ of $G$. Using careful refinements of the main steps of Tits’ proof of the alternative (which we will recall), we give sufficient conditions for the existence of ping-pong partners for the $H_i$ in any Zariski-dense subgroup $\Gamma$.

We will also show that these conditions are satisfied for products of copies of $\mathrm{SL}_n$ over division $\mathbb{R}$-algebras.

The existence of such free products has applications in the theory of integral group rings of finite groups, which will be briefly mentioned.

Joint work with G. Janssens and D. Temmerman.