Fixed points of endomorphisms of graph groups
  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. 
Emanuele Rodaro, Pedro V. Silva, Mihalis Sykiotis
math.GR