Random divergence of groups

2023/03/17 by Goldsborough, Antoine, Sisto, Alessandro · 1 citation
#FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2303.09943

Abstract

The divergence of a group is a quasi-isometry invariant defined in terms of pairs of points and lengths of paths avoiding a suitable ball around the identity. In this paper we study "random divergence'', meaning the divergence at two points chosen according to independent random walks or Markov chains; the Markov chains version can be turned into a quasi-isometry invariant. We show that in many cases, such as for relatively hyperbolic groups, mapping class groups, and right-angled Artin groups, the divergence at two randomly chosen points is with high probability equivalent to the divergence of the group. That is, generic points realise the largest possible divergence.

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• Growth gaps and exponential genericity in acylindrically hyperbolic groups

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