Singular value gap estimates for free products of semigroups

TL;DR

This paper provides lower bounds for singular value gaps in free products of certain semigroups and demonstrates that quasi-isometric embedding properties are preserved under free products of linear groups.

Contribution

It introduces new estimates for singular value gaps and shows that quasi-isometric embedding is preserved under free products of linear groups.

Findings

Lower bounds for singular value gaps established.

Quasi-isometric embedding is preserved in free products.

Class of linearly embeddable finitely generated groups is closed under free products.

Abstract

We establish lower estimates for singular value gaps of free products of $1$-divergent semigroups $\Gamma_1,\Gamma_2\subset \mathsf{GL}_d(\mathbb{K})$ which are in ping-pong position. As an application, we prove that if $\Gamma_1$ and $\Gamma_2$ are quasi-isometrically embedded subgroups in ping pong position, then the group they generate $\langle \Gamma_1,\Gamma_2\rangle$ is also quasi-isometrically embedded. In addition, we establish that the class of linear finitely generated groups, admitting a faithful linear representation over $\mathbb{R}$ which is a quasi-isometric embedding, is closed under free products.