Dirichlet domains for Anosov subgroups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916000436322304 |
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| author | Davalo, Colin Riestenberg, J. Maxwell |
| author_facet | Davalo, Colin Riestenberg, J. Maxwell |
| contents | We introduce a sufficient condition for a finitely generated subgroup $Γ$ of a semisimple Lie group $G$ to admit finite-sided Dirichlet domains for polyhedral Finsler metrics on the symmetric space $G/K$. The condition always implies the $Θ$-Anosov condition for some $Θ$, and can be arranged to be equivalent to the $Θ$-Anosov condition when $G$ is simple and $Θ$ is the set of long roots or the set of short roots. The Dirichlet domain we obtain extends to a fundamental domain for the action of $Γ$ on a domain of discontinuity in a flag manifold.
For instance, Borel Anosov subgroups of $\mathrm{SL}(d,\mathbb{R})$ have finite-sided Dirichlet domains for the Hilbert metric on the symmetric space which extends to the space of line-hyperplane flags, and $n$-Anosov subgroups of $\mathrm{Sp}(2n,\mathbb{R})$ have finite-sided Dirichlet-Selberg domains in $\mathrm{SL}(2n,\mathbb{R})/\mathrm{SO}(2n)$ which extend to a domain in projective space bounded by quadrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06408 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirichlet domains for Anosov subgroups Davalo, Colin Riestenberg, J. Maxwell Geometric Topology Group Theory 22E40, 53C35, 20F65 We introduce a sufficient condition for a finitely generated subgroup $Γ$ of a semisimple Lie group $G$ to admit finite-sided Dirichlet domains for polyhedral Finsler metrics on the symmetric space $G/K$. The condition always implies the $Θ$-Anosov condition for some $Θ$, and can be arranged to be equivalent to the $Θ$-Anosov condition when $G$ is simple and $Θ$ is the set of long roots or the set of short roots. The Dirichlet domain we obtain extends to a fundamental domain for the action of $Γ$ on a domain of discontinuity in a flag manifold. For instance, Borel Anosov subgroups of $\mathrm{SL}(d,\mathbb{R})$ have finite-sided Dirichlet domains for the Hilbert metric on the symmetric space which extends to the space of line-hyperplane flags, and $n$-Anosov subgroups of $\mathrm{Sp}(2n,\mathbb{R})$ have finite-sided Dirichlet-Selberg domains in $\mathrm{SL}(2n,\mathbb{R})/\mathrm{SO}(2n)$ which extend to a domain in projective space bounded by quadrics. |
| title | Dirichlet domains for Anosov subgroups |
| topic | Geometric Topology Group Theory 22E40, 53C35, 20F65 |
| url | https://arxiv.org/abs/2402.06408 |