Rigidity of warped cones and coarse geometry of expanders

2017/10/09 by David Fisher, Thang Nguyen, Fisher, David +3 · 3 citations
Mathematics · #Advanced Operator Algebra Research #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.1710.03085

openalex publication_date 2017/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the geometry of warped cones over free, minimal isometric group actions and related constructions of expander graphs. We prove a rigidity theorem for the coarse geometry of such warped cones: Namely, if a group has no abelian factors, then two such warped cones are quasi-isometric if and only if the actions are finite covers of conjugate actions. As a consequence, we produce continuous families of non-quasi-isometric expanders and superexpanders. The proof relies on the use of coarse topology for warped cones, such as a computation of their coarse fundamental groups.

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