Robust quasi-isometric embeddings of virtually free groups

TL;DR

This paper constructs new examples of robust quasi-isometric embeddings of free groups into general linear groups over various fields, which are not limits of Anosov representations, advancing understanding of geometric group embeddings.

Contribution

It provides the first known robust quasi-isometric embeddings of non-elementary free groups into $ ext{GL}_n(k)$ that are not limits of Anosov representations, including new examples over nonarchimedean and complex fields.

Findings

Constructed robust quasi-isometric embeddings into $ ext{GL}_n(k)$ for nonarchimedean fields.

Provided examples over $ ext{GL}_n(f{K})$ with $f{K}= eal,\complex$ that are not limits of Anosov representations.

Presented a non-Anosov embedding of the free semigroup into $ ext{GL}_3(f{C})$ as a limit of Anosov representations.

Abstract

Let $k$ be a nonarchimedean local field. For any $n\geq 3$, 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 $\bf{K}=\mathbb{R},\mathbb{C}$, we exhibit examples of non-locally rigid, robust quasi-isometric embeddings of virtually free groups into $\mathsf{GL}_n(\bf{K})$, $n\geq 3$, 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.