Random walks, WPD actions, and the Cremona group

Abstract

We study random walks on the Cremona group. We show that almost surely the dynamical degree of a sequence of random Cremona transformations grows exponentially fast, and a random walk produces infinitely many different normal subgroups with probability 1. Moreover, we study the structure of such random subgroups. We prove these results in general for groups of isometries of (non‐proper) hyperbolic spaces which possess at least one WPD element. As another application, we answer a question of Margalit showing that a random normal subgroup of the mapping class group is free.

Random walks and quasi‐convexity in acylindrically hyperbolic groups

“…Given a reversible, non-elementary, WPD probability distribution μ on G, there exists a unique, maximal finite subgroup of G normalized by Γ μ [39,Lemma 5.5]; see also [48,Proposition 1.14]. We denote this subgroup by E G (μ), or just E(μ) when G is understood.…”

“…Theorem 2.9 [48,Corollary 11.5]. The probability that w is loxodromic and WPD with E(w) = E + (w) = w tends to one as n tends to infinity.…”

“…We note that the term 'self-match' is used slightly differently in [48] where they require that p and p be disjoint. Proposition 2.12 [26, Proposition 1.5; 48, Proposition 7.5].…”

“…Thus there is a subpath of α of length at least (ε + ε − 1)Dn − 2K 0 whose image under h is contained in the (2K 0 + K)-neighborhood of α. Therefore there is an ε > 0 such that α has a (ε Dn, 2K 0 + K)-self-match for all sufficiently large n. As h ∈ w = E(w), the probability that this occurs approaches 0 as n → ∞ by [48,Proposition 11.7].…”

“…We use random walks to give a probabilistic generalization of these existence theorems which holds for the much larger class of acylindrically hyperbolic groups G and many subgroups H with quasi-convex orbits for an appropriate action of G. Random walks in groups acting on hyperbolic metric spaces have recently been studied by a number of people [20,[46][47][48]65]. In particular, Maher-Tiozzo showed (among many other things) that if G has a nonelementary action on a hyperbolic metric space X, then a random element of G will act loxodromically on X [47].…”