Brownian motion and orbit counting of Kleinian groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909988162633728 |
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| author | Hao, Yanlong Liu, Beibei |
| author_facet | Hao, Yanlong Liu, Beibei |
| contents | In this paper, we investigate the relationship between the divergence of Kleinian groups $Γ$ and the recurrence of simple random walks on the Schreier graph associated with $Γ$. In particular, we show that if $Γ$ is a subgroup of a lattice and is of divergence type, then the Schreier graph is recurrent. Our approach builds connections among the growth rate of the $Γ$-orbit, the volume growth rate of the quotient manifolds, and the growth rate of the Schreier graph. Using the connections, we construct abundant Kleinian groups of divergence type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Brownian motion and orbit counting of Kleinian groups Hao, Yanlong Liu, Beibei Geometric Topology Group Theory Probability 22E40, 58J50, 05C81 In this paper, we investigate the relationship between the divergence of Kleinian groups $Γ$ and the recurrence of simple random walks on the Schreier graph associated with $Γ$. In particular, we show that if $Γ$ is a subgroup of a lattice and is of divergence type, then the Schreier graph is recurrent. Our approach builds connections among the growth rate of the $Γ$-orbit, the volume growth rate of the quotient manifolds, and the growth rate of the Schreier graph. Using the connections, we construct abundant Kleinian groups of divergence type. |
| title | Brownian motion and orbit counting of Kleinian groups |
| topic | Geometric Topology Group Theory Probability 22E40, 58J50, 05C81 |
| url | https://arxiv.org/abs/2508.09807 |