Hierarchical hyperbolicity of graph products

2020/06/04 by Daniel Berlyne, Berlyne, Daniel, Jacob A. Russell +1 · 2 citations
Mathematics · #20F55 #20F65 #20F67 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2006.03085

openalex publication_date 2020/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. We also answer two questions of Genevois about the geometry of the electrification of a graph product of finite groups.

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• The metric Rips filtration, universal quasigeodesic cones, and hierarchically hyperbolic spaces
• Embedability of right-angled Artin groups into hierarchically hyperbolic groups

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