Random walks, WPD actions, and the Cremona group

Abstract

We study random walks on groups of isometries of non-proper delta-hyperbolic spaces under the assumption that at least one element in the group satisfies Bestvina-Fujiwara's WPD condition. We show that in this case typical elements are WPD, and the Poisson boundary coincides with the Gromov boundary. Moreover, we show that the random walk satisfies a form of asymptotic acylindricality, and we use this to show that the normal closure of random elements yields almost surely infinitely many different normal subgroups. Moreover, the probability that the normal closure is free tends to 1 if and only if the maximal normal subgroup coincides with the center of the group. We apply such techniques to the Cremona group, thus obtaining that the dynamical degree of random Cremona transformations grows exponentially fast, producing many different normal subgroups, and identifying the Poisson boundary. We also give a new identification of the Poisson boundary of Out(F_n). Our methods give bounds on the rates of convergence for these results.

arXiv comment54 pages, 8 figures. Various revisions; improved rate of decay to exponential

References in corpus (19)

• Geometry of the Complex of Curves I: Hyperbolicity
• Bounded cohomology of subgroups of mapping class groups
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Value distribution for sequences of rational mappings and complex dynamics
• Random walks on the mapping class group
• Normal subgroups in the Cremona group (long version)
• The free splitting complex of a free group I: Hyperbolicity
• Exponential decay in the mapping class group
• Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
• The co-surface graph and the geometry of hyperbolic free group extensions
• Normal subgroup generated by a plane polynomial automorphism
• Non simplicité du groupe de Cremona sur tout corps
• Kähler groups, real hyperbolic spaces and the Cremona group
• Statistics and compression of scl
• The Poisson boundary of $\text{Out}(F_N)$
• Partial periodic quotient of groups acting on a hyperbolic space
• Acylindrical group actions on quasi-trees
• Acylindrical hyperbolicity of the three-dimensional tame automorphism group
• Some effectivity questions for plane Cremona transformations

Cited by in corpus (3)

• Asymptotic translation lengths and normal generation for pseudo-Anosov monodromies of fibered 3-manifolds
• Random walks and quasi-convexity in acylindrically hyperbolic groups
• The Poisson boundary of lampshuffler groups