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

TL;DR

This work develops a unified framework for end-fixed subgroups in direct products of free or surface groups with a free-abelian factor. By leveraging the known endomorphism forms for $F_n\times\mathbb Z^m$ and $\pi_1(\Sigma_g)\times\mathbb Z^m$, the authors derive explicit structural descriptions of fixed subgroups and establish sharp obstructions showing these fixed subgroups cannot be infinite-rank free or exceed certain rank bounds. They prove Hopfian-but-not-co-Hopfian properties for $G=H\times\mathbb Z^m$ (with $H$ free or surface or hyperbolic) and extend end-fixed classifications from automorphisms to endomorphisms across these categories, including a complete classification in the free-abelian×surface setting and insights for hyperbolic cases via first Betti number considerations and finite-index rigidity. The results illuminate the landscape of fixed subgroups across free, surface, and hyperbolic groups, with implications for subgroup dynamics under endomorphisms and for understanding automorphism-fixed versus end-fixed phenomena in product 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.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Remark 1.8
• Lemma 2.1: Schreier's formula
• Lemma 2.2: Griffiths, Gr63