Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912527737159680 |
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| author | Lee, Homin Van LimBeek, Wouter Tiozzo, Giulio |
| author_facet | Lee, Homin Van LimBeek, Wouter Tiozzo, Giulio |
| contents | We consider symmetric random walks on discrete, Zariski-dense subgroups $Γ$ of a semisimple Lie group $G$ with Property (T). We prove that if $Γ$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06329 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank Lee, Homin Van LimBeek, Wouter Tiozzo, Giulio Dynamical Systems Group Theory Probability We consider symmetric random walks on discrete, Zariski-dense subgroups $Γ$ of a semisimple Lie group $G$ with Property (T). We prove that if $Γ$ has infinite covolume, then the associated hitting measure on the Furstenberg boundary of $G$ is singular. This is in contrast to Furstenberg's discretization of Brownian motion to lattices, and it is the first result of this type when $G$ has higher rank. |
| title | Singularity of Furstenberg measure for infinite covolume discrete subroups in higher rank |
| topic | Dynamical Systems Group Theory Probability |
| url | https://arxiv.org/abs/2508.06329 |