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<p>Let <i>G</i> be a countable group whose action on a metric space <i>X</i> involves a contracting isometry. This setting naturally encompasses groups acting on Gromov hyperbolic spaces, Teichmüller space, Culler-Vogtmann Outer space and CAT(0) spaces. We discuss continuity and differentiability of the escape rate of random walks on <i>G</i>. For relatively hyperbolic groups, CAT(-1) groups and CAT(0) cubical groups, we further discuss analyticity of the escape rate. Finally, assuming that the action of <i>G</i> on <i>X</i> is weakly properly discontinuous (WPD), we discuss continuity of the asymptotic entropy of random walks on <i>G</i>.</p>

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Contracting isometries and differentiability of the escape rate

  • 期刊论文

  • 发表日期：

    2025年10月17日

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  • Inhyeok Choi

摘要

Let G be a countable group whose action on a metric space X involves a contracting isometry. This setting naturally encompasses groups acting on Gromov hyperbolic spaces, Teichmüller space, Culler-Vogtmann Outer space and CAT(0) spaces. We discuss continuity and differentiability of the escape rate of random walks on G. For relatively hyperbolic groups, CAT(-1) groups and CAT(0) cubical groups, we further discuss analyticity of the escape rate. Finally, assuming that the action of G on X is weakly properly discontinuous (WPD), we discuss continuity of the asymptotic entropy of random walks on G.