Non-concentration property of Patterson-Sullivan measures for Anosov subgroups
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866910520650498048 |
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| author | Kim, Dongryul M. Oh, Hee |
| author_facet | Kim, Dongryul M. Oh, Hee |
| contents | Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $Γ<G$ with respect to a parabolic subgroup $P_θ$, we prove that any $Γ$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_θ$. More generally, we prove that for a Zariski dense $θ$-transverse subgroup $Γ<G$, any $(Γ, ψ)$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_θ$, provided the $ψ$-Poincaré series of $Γ$ diverges at $s=1$. In particular, our result also applies to relatively Anosov subgroups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_14911 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Non-concentration property of Patterson-Sullivan measures for Anosov subgroups Kim, Dongryul M. Oh, Hee Dynamical Systems Group Theory Geometric Topology Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $Γ<G$ with respect to a parabolic subgroup $P_θ$, we prove that any $Γ$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_θ$. More generally, we prove that for a Zariski dense $θ$-transverse subgroup $Γ<G$, any $(Γ, ψ)$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_θ$, provided the $ψ$-Poincaré series of $Γ$ diverges at $s=1$. In particular, our result also applies to relatively Anosov subgroups. |
| title | Non-concentration property of Patterson-Sullivan measures for Anosov subgroups |
| topic | Dynamical Systems Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2304.14911 |