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

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arxiv 2208.04778 v1 pith:PQL3ADMK submitted 2022-08-09 math.GR math.DS

Ilya Gekhtman , Yulan Qing , Kasra Rafi This is my paper · ORCID
classification math.GRmath.DS
keywords spacesublinearlyboundarymorsecontractingdirectionsgeodesicgroup
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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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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Sublinearly Morseness in Higher Rank Symmetric Spaces

    math.DG 2025-04 conditional novelty 7.0 of 10

    In higher-rank symmetric spaces, the new sublinearly Morse boundary has full Patterson-Sullivan measure for transverse groups, while the classical Morse boundary is negligible unless the group is Anosov.

  2. Atypical generic directions in Teichm\"uller space

    math.GT 2025-04 conditional novelty 6.0 of 10

    Explicit Teichmüller geodesic rays exist that are sublinearly Morse yet have minimal non-uniquely ergodic vertical foliations, giving the first such examples.

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