Fixed subgroups are compressed in surface groups

Abstract

For a compact surface $\Sigma$ (orientable or not, and with boundary or not) we show that the fixed subgroup, $\mathrm{Fix}B$, of any family $B$ of endomorphisms of ${\pi }_1(\Sigma )$ is compressed in ${\pi }_1(\Sigma )$ i.e., $\mathrm{rk}((\mathrm{Fix}B)\cap H)\le \mathrm{rk}(H)$ for any subgroup $\mathrm{Fix}B\le H\le {\pi }_1(\Sigma )$. 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 $\mathrm{rk}(\mathrm{Fix}\varphi )\le \mathrm{rk}(G)$ for every $\varphi \in Aut(G)$.