Approximating hyperbolic lattices by cubulations
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913389532413952 |
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| author | Brody, Nic Reyes, Eduardo |
| author_facet | Brody, Nic Reyes, Eduardo |
| contents | We show that an isometric action of a torsion-free uniform lattice $Γ$ on hyperbolic space $\mathbb{H}^n$ can be metrically approximated by geometric actions of $Γ$ on $\mathrm{CAT}(0)$ cube complexes, provided that either $n$ is at most three, or the lattice is arithmetic of simplest type. This solves a conjecture of Futer and Wise.
Our main tool is the study of a space of co-geodesic currents, consisting of invariant Radon measures supported on codimension-1 hyperspheres in the Gromov boundary of $\mathbb{H}^n$. By pairing co-geodesic currents and geodesic currents via an intersection number, we show that asymptotic convergence of geometric actions can be deduced from the convergence of their dual co-geodesic currents.
For surface groups, our methods also imply approximation by cubulations for actions induced by non-positively curved Riemannian surfaces with singularities, Hitchin and maximal representations, and quasiFuchsian representations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01511 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximating hyperbolic lattices by cubulations Brody, Nic Reyes, Eduardo Group Theory Geometric Topology We show that an isometric action of a torsion-free uniform lattice $Γ$ on hyperbolic space $\mathbb{H}^n$ can be metrically approximated by geometric actions of $Γ$ on $\mathrm{CAT}(0)$ cube complexes, provided that either $n$ is at most three, or the lattice is arithmetic of simplest type. This solves a conjecture of Futer and Wise. Our main tool is the study of a space of co-geodesic currents, consisting of invariant Radon measures supported on codimension-1 hyperspheres in the Gromov boundary of $\mathbb{H}^n$. By pairing co-geodesic currents and geodesic currents via an intersection number, we show that asymptotic convergence of geometric actions can be deduced from the convergence of their dual co-geodesic currents. For surface groups, our methods also imply approximation by cubulations for actions induced by non-positively curved Riemannian surfaces with singularities, Hitchin and maximal representations, and quasiFuchsian representations. |
| title | Approximating hyperbolic lattices by cubulations |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2404.01511 |