Random walks, WPD actions, and the Cremona group

2018/07/31 by Joseph Maher, Giulio Tiozzo · 7 citations
Mathematics · #Class (philosophy) #Degree (music) #Geometric and Algebraic Topology #Group (periodic table) #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Random graph #Random walk #Sequence (biology) #math.AG #math.DS #math.GT #math.PR #msc:20F67 #msc:57M60 #msc:60G50

paper · pdf · doi:10.1112/plms.12394

published in Proceedings of the London Mathematical Society 123(2), 153-202 (Wiley) · 54 pages, 8 figures. Various revisions; improved rate of decay to exponential

openalex created_date 2018/08/03 · openalex publication_date 2021/01/04 · arxiv created 2021/01/05 · arxiv updated 2021/01/13 · openalex updated_date 2026/08/05

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.

Citations

• Géométrie et théorie des groupes
• Some effectivity questions for plane Cremona transformations
• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Geometry of the complex of curves I: Hyperbolicity
• Groups Covered By Permutable Subsets
• Tight geodesics in the curve complex
• Relatively Hyperbolic Groups
• Acylindrical accessibility for groups
• The Poisson Formula for Groups with Hyperbolic Properties
• The co‐surface graph and the geometry of hyperbolic free group extensions
• Acylindrically hyperbolic groups
• Hyperbolic Groups
• Embeddings of Gromov hyperbolic spaces

Cited by

• Non-solutions to mixed equations in acylindrically hyperbolic groups coming from random walks
• Normal generators for mapping class groups