Counting loxodromics for hyperbolic actions

Abstract

Let $G\curvearrowright X$ be a nonelementary action by isometries of a hyperbolic group $G$ on a hyperbolic metric space $X$. We show that the set of elements of $G$ which act as loxodromic isometries of $X$ is generic. That is, for any finite generating set of $G$, the proportion of $X$--loxodromics in the ball of radius $n$ about the identity in $G$ approaches $1$ as $n\to \infty$. We also establish several results about the behavior in $X$ of the images of typical geodesic rays in $G$; for example, we prove that they make linear progress in $X$ and converge to the Gromov boundary $\partial X$. Our techniques make use of the automatic structure of $G$, Patterson--Sullivan measure on $\partial G$, and the ergodic theory of random walks for groups acting on hyperbolic spaces. We discuss various applications, in particular to Mod(S), Out($F_N$), and right--angled Artin groups.

References in corpus (12)

• Geometry of the Complex of Curves I: Hyperbolicity
• Random walks on the mapping class group
• Finite covers of random 3-manifolds
• Amenable hyperbolic groups
• Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
• Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups
• The co-surface graph and the geometry of hyperbolic free group extensions
• On hyperbolicity of free splitting and free factor complexes
• Statistics and compression of scl
• Generalized north-south dynamics on the space of geodesic currents
• Subgroup decomposition in Out(F_n): Introduction and Research Announcement
• Linear Progress with Exponential Decay in Weakly Hyperbolic Groups

Cited by in corpus (4)

• Pseudo-Anosovs are exponentially generic in mapping class groups
• The Manhattan curve, ergodic theory of topological flows and rigidity
• Topological flows for hyperbolic groups
• Counting pseudo-Anosovs as weakly contracting isometries