Random quotients preserve acylindrical and hierarchical hyperbolicity

2025/07/22 by Abbott, Carolyn, Berlyne, Daniel, Mangioni, Giorgio +2
#05C81 (Secondary) #20F65 (Primary) 20F67 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2507.16677

Abstract

We show that random quotients of acylindrically hyperbolic groups, obtained by taking a quotient of the group by the nth steps of a finite collection of independent random walks, are again acylindrically hyperbolic asymptotically almost surely. Our main tools come from spinning families and projection complexes, which we relate to random walks and develop further. Furthermore, we show that a random quotient of a hierarchically hyperbolic group is again hierarchically hyperbolic asymptotically almost surely. The same techniques also yield that a random quotient of a non-elementary hyperbolic group (relative to a finite collection of peripheral subgroups) is asymptotically almost surely hyperbolic (relative to isomorphic peripheral subgroups).

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• Tracking rates of random walks
• Random quotients of mapping class groups are quasi-isometrically rigid
• Random walk on hyperbolic groups and proper powers
• Subgroup mixing and random walks in groups acting on hyperbolic spaces