Classification of aut-fixed subgroups in free-abelian times surface groups

TL;DR

This paper classifies fixed subgroups of automorphisms in groups formed by the product of surface groups and free abelian groups, revealing new structural insights and applications to Nielsen fixed point theory.

Contribution

It provides a complete classification of fixed subgroups in aut-fixed subgroups of free-abelian times surface groups, highlighting differences from hyperbolic groups.

Findings

Infinitely many fixed subgroups exist if and only if k ≥ 2.

Constructs examples of aspherical manifolds lacking Jiang's Bound Index Property.

Provides results on fixed subgroups in hyperbolic group products.

Abstract

In this paper, we are concerned with the direct product $G=\pi_1(\Sigma)\times \Z^k$ for $\Sigma$ a compact orientable surface with negative Euler characteristic, and give a complete classification of its fixed subgroups of automorphisms. As a corollary, we show that $G$ contains, up to isomorphism, infinitely many fixed subgroups of automorphisms if and only if $k\geq 2$, which is a contrast to that of hyperbolic groups. As an application on Nielsen fixed point theory, we provide a family of aspherical manifolds without Jiang's Bound Index Property. Moreover, we also give some results on the fixed subgroups of the direct product $H\times \Z^k$ for $H$ a non-elementary torsion-free hyperbolic group.