The group fixed by a family of endomorphisms of a surface group

Abstract

For a closed surface $S$ with $\chi(S)<0$, we show that the fixed subgroup of a family $\mathcal B$ of endomorphisms of $\pi_1(S)$ has $\rk \fix\mathcal B\leq \rk \pi_1(S)$. In particular, if $\mathcal B$ contains a non-epimorphic endomorphism, then $\rk \fix\mathcal B\leq \frac{1}{2} \rk \pi_1(S)$. We also show that geometric subgroups of $\pi_1(S)$ are inert, and hence the fixed subgroup of a family of epimorphisms of $\pi_1(S)$ is also inert.