Random walks on cocompact Fuchsian and Kleinian groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915666730156032 |
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| author | Bogachev, Nikolay Kosenko, Peter Tiozzo, Giulio |
| author_facet | Bogachev, Nikolay Kosenko, Peter Tiozzo, Giulio |
| contents | The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup $Γ$ of $\mathrm{SL}_N(\mathbb R)$ is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on ``most'' cocompact Fuchsian and Kleinian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09900 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random walks on cocompact Fuchsian and Kleinian groups Bogachev, Nikolay Kosenko, Peter Tiozzo, Giulio Dynamical Systems Group Theory Geometric Topology Probability The question of the singularity at infinity of the hitting measure of random walks has a long history, originating from the work of Furstenberg in the 1960s. In 2011, Kaimanovich and Le Prince conjectured that the hitting measure of any finitely supported random walk on a discrete subgroup $Γ$ of $\mathrm{SL}_N(\mathbb R)$ is singular at infinity with respect to the Lebesgue measure. Using algebraic and geometric convergence and hyperbolic Dehn filling, we prove the singularity conjecture for certain measures on ``most'' cocompact Fuchsian and Kleinian groups. |
| title | Random walks on cocompact Fuchsian and Kleinian groups |
| topic | Dynamical Systems Group Theory Geometric Topology Probability |
| url | https://arxiv.org/abs/2512.09900 |