Classification of Fixed Subgroups of Endomorphisms in Free-abelian Times Surface Groups

TL;DR

This paper characterizes endomorphisms of free-abelian times surface groups, classifies their fixed subgroups, and explores properties of related hyperbolic groups, advancing understanding of their algebraic structure.

Contribution

It provides a complete classification of fixed subgroups of endomorphisms in free-abelian times surface groups, extending previous results for automorphisms.

Findings

Free-abelian times hyperbolic groups are Hopfian but not co-Hopfian.

Complete classification of fixed subgroups of endomorphisms in free-abelian times surface groups.

Conditions for finitely many end-fixed subgroups in free-abelian times hyperbolic groups.

Abstract

In this paper, we first study the endomorphisms of free-abelian times surface groups and give a characterization of when they are injective and surjective. Then, we see that free-abelian times hyperbolic groups are Hopfian but not co-Hopfian. Moreover, we give a complete classification of fixed subgroups of endomorphisms in free-abelian times surface groups, which extends that of automorphisms. Finally, we study the endomorphisms of free-abelian times non-elementary torsion-free hyperbolic groups and give an equivalent condition for them to contain, up to isomorphism, finitely many end-fixed subgroups.