Conformal dynamics at infinity for groups with contracting elements

TL;DR

This paper develops a unified theory of conformal density at infinity for groups with contracting elements, encompassing various known boundaries and providing new applications in geometric group theory.

Contribution

It introduces a convergence boundary framework and establishes conformal density theory, unifying multiple boundary theories and extending applications to random walks and group co-growth.

Findings

Unified conformal density theory for hyperbolic-like boundaries

Identification of Poisson boundaries for random walks

New applications in CAT(0) and mapping class groups

Abstract

This paper develops a theory of conformal density at infinity for groups with contracting elements. We start by introducing a class of convergence boundary encompassing many known hyperbolic-like boundaries, on which a detailed study of conical points and Myrberg points is carried out. The basic theory of conformal density is then established on the convergence boundary, including the Sullivan shadow lemma and a Hopf--Tsuji--Sullivan dichotomy. This gives a unification of the theory of conformal density on the Gromov and Floyd boundary for (relatively) hyperbolic groups, the visual boundary for rank-1 CAT(0) groups, and Thurston boundary for mapping class groups. Besides that, the conformal density on the horofunction boundary provides a new important example of our general theory. Applications include the identification of Poisson boundary of random walks, the co-growth problem of divergent groups, measure theoretical results for CAT(0) groups and mapping class groups.