For finitely presented acylindrically hyperbolic groups with at most polynomial Dehn function, the random Dehn function is at most quadratic and strictly smaller than the classical Dehn function when the group is not hyperbolic.
Random divergence of groups
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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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Random Dehn function of groups
For finitely presented acylindrically hyperbolic groups with at most polynomial Dehn function, the random Dehn function is at most quadratic and strictly smaller than the classical Dehn function when the group is not hyperbolic.