Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichm\"uller space

TL;DR

This paper demonstrates that sublinearly Morse directions are prevalent in the boundaries of rank-1 CAT(0) spaces and Teichmüller space, linking geometric boundary properties with probabilistic models like the Poisson boundary.

Contribution

It establishes the genericity of sublinearly Morse directions in CAT(0) and Teichmüller spaces and connects these directions to the Poisson boundary for random walks.

Findings

Sublinearly Morse directions are generic in CAT(0) and Teichmüller spaces.

Sublinearly Morse boundary models the Poisson boundary for certain groups.

A criterion for sublinear contraction of geodesic rays is developed.

Abstract

We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the word, namely with respect to Patterson-Sullivan measures and stationary measures for random walks. We deduce that the sublinearly Morse boundary is a model of the Poisson boundary for finitely supported random walks on groups acting geometrically on rank-1 CAT (0) spaces. We prove an analogous result for mapping class group actions on Teichm\"uller space. Our main technical tool is a criterion, valid in any unique geodesic metric space, that says that any geodesic ray with sufficiently many (in a statistical sense) strongly contracting segments is sublinearly contracting.