Conformal measure rigidity for representations via self-joinings
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914976927580160 |
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| author | Kim, Dongryul M. Oh, Hee |
| author_facet | Kim, Dongryul M. Oh, Hee |
| contents | Let $Γ$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $ρ:Γ\to G_2$ and a surprising role played by higher rank conformal measures of the associated self-joining group. Our approach recovers rigidity theorems of Sullivan, Tukia and Yue, as well as applies to Anosov representations, including Hitchin representations. More precisely, for a given representation $ρ$ with a boundary map $f$ defined on the limit set $Λ$, we ask whether the extendability of $ρ$ to $G$ can be detected by the property that $f$ pushes forward some $Γ$-conformal measure class $[ν_Γ]$ to a $ρ(Γ)$-conformal measure class $[ν_{ρ(Γ)}]$. When $Γ$ is of divergence type in a rank one group or when $ρ$ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining $Γ_ρ=(\text{id} \times ρ)(Γ)$ is Zariski dense in $G_1\times G_2$, then the push-forward measures $(\text{id}\times f)_*ν_Γ$ and $(f^{-1}\times \text{id})_*ν_{ρ(Γ)}$, which are higher rank $Γ_ρ$-conformal measures, cannot be in the same measure class. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_03539 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Conformal measure rigidity for representations via self-joinings Kim, Dongryul M. Oh, Hee Geometric Topology Dynamical Systems Group Theory Let $Γ$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $ρ:Γ\to G_2$ and a surprising role played by higher rank conformal measures of the associated self-joining group. Our approach recovers rigidity theorems of Sullivan, Tukia and Yue, as well as applies to Anosov representations, including Hitchin representations. More precisely, for a given representation $ρ$ with a boundary map $f$ defined on the limit set $Λ$, we ask whether the extendability of $ρ$ to $G$ can be detected by the property that $f$ pushes forward some $Γ$-conformal measure class $[ν_Γ]$ to a $ρ(Γ)$-conformal measure class $[ν_{ρ(Γ)}]$. When $Γ$ is of divergence type in a rank one group or when $ρ$ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining $Γ_ρ=(\text{id} \times ρ)(Γ)$ is Zariski dense in $G_1\times G_2$, then the push-forward measures $(\text{id}\times f)_*ν_Γ$ and $(f^{-1}\times \text{id})_*ν_{ρ(Γ)}$, which are higher rank $Γ_ρ$-conformal measures, cannot be in the same measure class. |
| title | Conformal measure rigidity for representations via self-joinings |
| topic | Geometric Topology Dynamical Systems Group Theory |
| url | https://arxiv.org/abs/2302.03539 |