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<p>We show that the sublinearly Morse directions in a visual boundary of a rank-1 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\operatorname {CAT}(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>CAT</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> group 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 the associated rank-1 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\operatorname {CAT}(0)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>CAT</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> spaces. We prove an analogous result for mapping class group actions on Teichmüller 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.</p>

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Genericity of sublinearly Morse directions in CAT(0) spaces and the Teichmüller space

  • 期刊论文

  • 发表日期：

    2025年10月8日

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  • Ilya Gekhtman,
  • Yulan Qing,
  • Kasra Rafi

摘要

We show that the sublinearly Morse directions in a visual boundary of a rank-1 \(\operatorname {CAT}(0)\) CAT ( 0 ) group 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 the associated rank-1 \(\operatorname {CAT}(0)\) CAT ( 0 ) spaces. We prove an analogous result for mapping class group actions on Teichmüller 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.