Fixed subgroups are compressed in surface groups

TL;DR

This paper proves that fixed subgroups of endomorphisms in surface groups are compressed, and explores related conjectures and properties in free and surface groups, including direct products.

Contribution

It establishes the compression property of fixed subgroups in surface groups and provides partial solutions to the inertia conjecture for free and surface groups.

Findings

Fixed subgroups are compressed in surface groups.

Partial positive solution to the inertia conjecture.

Characterization of automorphisms in direct products of free and surface groups.

Abstract

For a compact surface $\Sigma$ (orientable or not, and with boundary or not) we show that the fixed subgroup, $\operatorname{Fix} B$, of any family $B$ of endomorphisms of $\pi_1(\Sigma)$ is compressed in $\pi_1(\Sigma)$ i.e., $\operatorname{rk}((\operatorname{Fix} B)\cap H)\leq \operatorname{rk}(H)$ for any subgroup $\operatorname{Fix} B \leq H \leq \pi_1(\Sigma)$. On the way, we give a partial positive solution to the inertia conjecture, both for free and for surface groups. We also investigate direct products, $G$, of finitely many free and surface groups, and give a characterization of when $G$ satisfies that $\operatorname{rk}(\operatorname{Fix} \phi) \leq \operatorname{rk}(G)$ for every $\phi \in Aut(G)$.