$\mathbb{H}^{p,q}$-convex cocompactness and higher higher Teichmüller spaces
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914071891148800 |
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| author | Beyrer, Jonas Kassel, Fanny |
| author_facet | Beyrer, Jonas Kassel, Fanny |
| contents | For any integers $p\geq 2$ and $q\geq 1$, let $\mathbb{H}^{p,q}$ be the pseudo-Riemannian hyperbolic space of signature $(p,q)$. We prove that if $Γ$ is the fundamental group of a closed aspherical $p$-manifold, then the set of representations of $Γ$ to $\mathrm{PO}(p,q+1)$ which are convex cocompact in $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(Γ,\mathrm{PO}(p,q+1))$. More generally, we show that if $Γ$ is any finitely generated group with no infinite nilpotent normal subgroups and with virtual cohomological dimension $p$, then the set of injective and discrete representations of $Γ$ to $\mathrm{PO}(p,q+1)$ preserving a non-degenerate non-positive $(p-1)$-sphere in the boundary of $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(Γ,\mathrm{PO}(p,q+1))$. This gives new examples of higher-dimensional higher-rank Teichmüller spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_15031 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $\mathbb{H}^{p,q}$-convex cocompactness and higher higher Teichmüller spaces Beyrer, Jonas Kassel, Fanny Geometric Topology Group Theory For any integers $p\geq 2$ and $q\geq 1$, let $\mathbb{H}^{p,q}$ be the pseudo-Riemannian hyperbolic space of signature $(p,q)$. We prove that if $Γ$ is the fundamental group of a closed aspherical $p$-manifold, then the set of representations of $Γ$ to $\mathrm{PO}(p,q+1)$ which are convex cocompact in $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(Γ,\mathrm{PO}(p,q+1))$. More generally, we show that if $Γ$ is any finitely generated group with no infinite nilpotent normal subgroups and with virtual cohomological dimension $p$, then the set of injective and discrete representations of $Γ$ to $\mathrm{PO}(p,q+1)$ preserving a non-degenerate non-positive $(p-1)$-sphere in the boundary of $\mathbb{H}^{p,q}$ is a union of connected components of $\mathrm{Hom}(Γ,\mathrm{PO}(p,q+1))$. This gives new examples of higher-dimensional higher-rank Teichmüller spaces. |
| title | $\mathbb{H}^{p,q}$-convex cocompactness and higher higher Teichmüller spaces |
| topic | Geometric Topology Group Theory |
| url | https://arxiv.org/abs/2305.15031 |