Eventually fixed points of endomorphisms of virtually free groups

Abstract

We consider the subgroup of points of finite orbit through the action of an endomorphism of a finitely generated virtually free group, with particular emphasis on the subgroup of eventually fixed points, $\text{EvFix}(\varphi)$: points whose orbit contains a fixed point. We provide an algorithm to compute the subgroup of fixed points of an endomorphism of a finitely generated virtually free group and prove that finite orbits have cardinality bounded by a computable constant, which allows us to solve several algorithmic problems: deciding if φ is a finite order element of $\text{End}(G)$, if φ is aperiodic, if $\text{EvFix}(\varphi)$ is finitely generated and if $\text{EvFix}(\varphi)$ is a normal subgroup. In the cases where $\text{EvFix}(\varphi)$ is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.