Robust quasi-isometric embeddings of virtually free groups

Abstract

Let $k$ be a nonarchimedean local field. For any $n\ge 3$, we construct the first examples of robust quasi-isometric embeddings of non-elementary free groups into ${\mathsf{GL}}_n(k)$ which are not limits of Anosov representations. If $\mathbf{K}=\mathbb{R},\mathbb{C}$, we exhibit examples of non-locally rigid, robust quasi-isometric embeddings of virtually free groups into ${\mathsf{GL}}_n(\mathbf{K})$, $n\ge 3$, which are not limits of Anosov representations. Moreover, we exhibit a non-Anosov robust quasi-isometric embedding of the free semigroup $\mathbb{Z}\ast {\mathbb{Z}}^{+}$ into ${\mathsf{GL}}_3(\mathbb{C})$, which is a limit of Anosov representations.

20 pages. Comments welcome