Klein-Maskit combination theorem for Anosov subgroups: Free products

TL;DR

This paper generalizes the Klein-Maskit combination theorem to the setting of Anosov subgroups, showing that under certain conditions, the free product of two Anosov subgroups remains Anosov, extending previous conjectures.

Contribution

It proves that the free product of two Anosov subgroups is again Anosov under suitable conditions, generalizing classical combination theorems.

Findings

The free product of two Anosov subgroups is Anosov.

The resulting group is isomorphic to the free product of the original subgroups.

The theorem confirms a previous conjecture about Anosov subgroup combinations.

Abstract

We prove a generalization of the classical Klein-Maskit combination theorem, in the free product case, in the setting of Anosov subgroups. Namely, if $\Gamma_A$ and $\Gamma_B$ are Anosov subgroups of a semisimple Lie group $G$ of noncompact type, then under suitable topological assumptions, the group generated by $\Gamma_A$ and $\Gamma_B$ in $G$ is again Anosov, and is naturally isomorphic to the free product $\Gamma_A*\Gamma_B$. Such a generalization was conjectured in our previous article with Bernhard Leeb (arXiv:1805.07374).