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 $\rk \fix\mathcal B\leq \rk π_1(S)$. In particular, if $\mathcal{B}$ contains a non-epimorphic endomorphism, then $\rk \fix\mathcal B\leq \frac{1}{2} \rk π_1(S)$. 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.

16 pages

References in corpus (1)

• Intersections of conjugates of Magnus subgroups of one-relator groups

Cited by in corpus (3)

• Explicit bounds for fixed subgroups of endomorphisms of free products
• The fixed subgroups of homeomorphisms of Seifert manifolds
• Fixed subgroups in direct products of surface groups of Euclidean type