Drilling hyperbolic groups

2024/06/20 by Daniel Groves, Groves, Daniel, Peter Haı̈ssinsky +9 · 1 citation
Mathematics · #20F65 #20F67 #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.2406.14667

openalex publication_date 2024/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Given a hyperbolic group G and a maximal infinite cyclic subgroup ⟨ g ⟩, we define a \it drilling of G along g, which is a relatively hyperbolic group pair (\widehatG, P). This is inspired by the well-studied procedure of drilling a hyperbolic 3--manifold along an embedded geodesic. We prove that, under suitable conditions, a hyperbolic group with 2-sphere boundary admits a drilling where the resulting relatively hyperbolic group pair (\widehatG, P) has relatively hyperbolic boundary S². This allows us to reduce the Cannon Conjecture (in the residually finite case) to a relative version, which is likely to be more tractable.

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