The study of the subgroup fixed by an automorphism of a free group started with the work of Gersten and Cooper in the late 80's, who proved that this subgroup must always be finitely generated, and has since then presented many interesting challenges. The problem of computing such a subgroup was settled by Bogopolski and Maslakova in 2016. These results have been generalized to any endomorphism of the free group: the first by Imrich and Turner in 1989 and the latter by Mutanguha in 2021.
In this talk, we will show Mutanguha's result can be extended to endomorphisms of virtually free groups. We will also consider the subgroup of points eventually fixed by an endomorphism of a virtually free group, which consists of the points whose orbit contains a fixed point. Among other general properties, we will see that it is decidable whether or not this subgroup is finitely generated (and in case it is, compute a basis for it) by computing a bound on the cardinality of finite orbits.