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arXiv · 2208.04778

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

Ilya Gekhtman·Yulan Qing·Kasra Rafi
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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.

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Ilya Gekhtman, Yulan Qing, Kasra Rafi. 2022-08-09. Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichm\"uller space. https://arxiv.org/abs/2208.04778

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