Surface groups among cubulated hyperbolic and one-relator groups

2024/06/04 by Henry Wilton, Wilton, Henry · 2 citations
Computer Science · Mathematics · #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2406.02121

openalex publication_date 2024/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a non-positively curved cube complex with hyperbolic fundamental group. We prove that π₁(X) has a non-free subgroup of infinite index unless π₁(X) is either free or a surface group, answering questions of Gromov and Whyte (in a special case) and Wise. A similar result for one-relator groups follows, answering a question posed by several authors. The proof relies on a careful analysis of free and cyclic splittings of cubulated groups.

Cited by

• Subgroups of Cyclically Amalgamated Free Products
• Quasi-convex surface subgroups in some one-relator groups with torsion

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