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

2023/09/24 by Jialin Lei, Peng Wang, Lei, Jialin +3 · 1 citation
Mathematics · #20F34 #20F65 #57M07 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2309.13540

openalex publication_date 2023/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we are concerned with the direct product G=π₁(Σ)× \Z^k for Σ 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≥ 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× \Z^k for H a non-elementary torsion-free hyperbolic group.

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• On Jiang's Bounded Index Property for products of nilmanifolds

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