Geometry and arithmetic of semi-arithmetic Fuchsian groups
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866909564200288256 |
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| author | Belolipetsky, Mikhail Cosac, Gregory Dória, Cayo Paula, Gisele Teixeira |
| author_facet | Belolipetsky, Mikhail Cosac, Gregory Dória, Cayo Paula, Gisele Teixeira |
| contents | Semi-arithmetic Fuchsian groups is a wide class of discrete groups of isometries of the hyperbolic plane which includes arithmetic Fuchsian groups, hyperbolic triangle groups, groups admitting a modular embedding, and others. We introduce a new geometric invariant of a semi-arithmetic group called stretch. Its definition is based on the notion of the Riemannian center of mass developed by Karcher and collaborators. We show that there exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch and coarea. The proof of this result uses the arithmetic Margulis lemma. We also show that when stretch is not bounded there exist infinite sequences of such groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_13151 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Geometry and arithmetic of semi-arithmetic Fuchsian groups Belolipetsky, Mikhail Cosac, Gregory Dória, Cayo Paula, Gisele Teixeira Group Theory Geometric Topology Metric Geometry Number Theory 20H10, 11F06, 30F10 Semi-arithmetic Fuchsian groups is a wide class of discrete groups of isometries of the hyperbolic plane which includes arithmetic Fuchsian groups, hyperbolic triangle groups, groups admitting a modular embedding, and others. We introduce a new geometric invariant of a semi-arithmetic group called stretch. Its definition is based on the notion of the Riemannian center of mass developed by Karcher and collaborators. We show that there exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch and coarea. The proof of this result uses the arithmetic Margulis lemma. We also show that when stretch is not bounded there exist infinite sequences of such groups. |
| title | Geometry and arithmetic of semi-arithmetic Fuchsian groups |
| topic | Group Theory Geometric Topology Metric Geometry Number Theory 20H10, 11F06, 30F10 |
| url | https://arxiv.org/abs/2310.13151 |