Spinning and random quotients preserve hierarchical hyperbolicity

Abstract: In this paper we consider random quotients of hierarchically hyperbolic groups, obtained by taking the quotient of the group by the n-th steps of a finite family of independent random walks. 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). Our main tools come from the theory of spinning families and projection complexes, which we further develop.