Eventually fixed points of endomorphisms of virtually free groups

TL;DR

This paper studies the subgroup of eventually fixed points under endomorphisms of virtually free groups, providing algorithms for computing fixed points, bounds on orbit sizes, and solutions to related decision problems.

Contribution

It introduces an algorithm to compute fixed points of endomorphisms in virtually free groups and establishes bounds on orbit sizes, enabling solutions to several algorithmic problems.

Findings

Finite orbits have a computable bounded cardinality

Algorithms for deciding if an endomorphism has finite order or is aperiodic

Bounds for the rank of the subgroup of eventually fixed points

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.