Fixed points of endomorphisms of graph groups

TL;DR

This paper characterizes when fixed point subgroups of endomorphisms in graph groups are finitely generated, linking this property to the group's structure as a free product of free abelian groups and extending results to automorphisms under certain conditions.

Contribution

It provides a complete characterization of graph groups where fixed point subgroups of all endomorphisms are finitely generated, including automorphisms in specific graph classes.

Findings

Fix$\,mbda$ is finitely generated iff G is a free product of free abelian groups.

The same holds for periodic points.

Results extend to automorphisms when the dependence graph is a transitive forest.

Abstract

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for the subgroup of periodic points. Similar results are obtained for automorphisms, if the dependence graph of $G$ is a transitive forest.