Zariski dense non-tempered subgroups in higher rank of nearly optimal growth
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| Format: | Preprint |
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2024
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| _version_ | 1866916786244419584 |
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| author | Fraczyk, Mikolaj Oh, Hee |
| author_facet | Fraczyk, Mikolaj Oh, Hee |
| contents | We construct the first example of a Zariski-dense, discrete, non-lattice subgroup $Γ_0$ of a higher rank simple Lie group $G$, which is non-tempered in the sense that the quasi-regular representation $L^2(Γ_0\backslash G)$ is non-tempered. More precisely, let $n\ge 3$ and let $Γ$ be the fundamental group of a closed hyperbolic $n$-manifold that contains a properly embedded totally geodesic hyperplane. We show that there exists a non-empty open subset $\mathcal O$ of $\operatorname{Hom}(Γ, \operatorname{SO}(n,2))$ such that for any $σ\in \mathcal O$, the subgroup $σ(Γ)$ is a Zariski-dense and non-tempered Anosov subgroup of $\operatorname{SO}(n,2)$. In addition, the growth indicator of $σ(Γ)$ is nearly optimal: it almost realizes the supremum of growth indicators among all non-lattice discrete subgroups, a bound imposed by property $(T)$ of $\operatorname{SO}(n,2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_19551 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Zariski dense non-tempered subgroups in higher rank of nearly optimal growth Fraczyk, Mikolaj Oh, Hee Group Theory Dynamical Systems Geometric Topology Representation Theory Spectral Theory We construct the first example of a Zariski-dense, discrete, non-lattice subgroup $Γ_0$ of a higher rank simple Lie group $G$, which is non-tempered in the sense that the quasi-regular representation $L^2(Γ_0\backslash G)$ is non-tempered. More precisely, let $n\ge 3$ and let $Γ$ be the fundamental group of a closed hyperbolic $n$-manifold that contains a properly embedded totally geodesic hyperplane. We show that there exists a non-empty open subset $\mathcal O$ of $\operatorname{Hom}(Γ, \operatorname{SO}(n,2))$ such that for any $σ\in \mathcal O$, the subgroup $σ(Γ)$ is a Zariski-dense and non-tempered Anosov subgroup of $\operatorname{SO}(n,2)$. In addition, the growth indicator of $σ(Γ)$ is nearly optimal: it almost realizes the supremum of growth indicators among all non-lattice discrete subgroups, a bound imposed by property $(T)$ of $\operatorname{SO}(n,2)$. |
| title | Zariski dense non-tempered subgroups in higher rank of nearly optimal growth |
| topic | Group Theory Dynamical Systems Geometric Topology Representation Theory Spectral Theory |
| url | https://arxiv.org/abs/2410.19551 |