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 virtually free group, with particular emphasis on the subgroup of eventually fixed points, 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 $\varphi$ is a finite order element of End($G$), if $\varphi$ is aperiodic, if EvFix($\varphi$) is finitely generated and, in the free group case, whether EvFix($\varphi$) is a normal subgroup of $F_n$ or not. We also present a bound for the rank of EvFix($\varphi$) in case it is finitely generated.