Geodesic
The fundamental inequality for cocompact Fuchsian groups
Forum of Mathematics, Sigma, Tome 10 (2022)

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We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups. 
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     author = {Petr Kosenko and Giulio Tiozzo},
     title = {The fundamental inequality for cocompact {Fuchsian} groups},
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Petr Kosenko; Giulio Tiozzo. The fundamental inequality for cocompact Fuchsian groups. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.94

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