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, EvFix (): 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 End (G), if φ is aperiodic, if EvFix () is finitely generated and if EvFix () is a normal subgroup. In the cases where EvFix () is finitely generated, we also present a bound for its rank and an algorithm to compute a generating set.