Finiteness of CAT(0) group actions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913334744317952 |
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| author | Cavallucci, Nicola Sambusetti, Andrea |
| author_facet | Cavallucci, Nicola Sambusetti, Andrea |
| contents | We prove some finiteness results for discrete isometry groups $Γ$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M = Γ\backslash X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $Γ$ is bounded above by a universal constant only depending on the packing constants and the codiameter. The main tool is a splitting theorem for sufficiently collapsed actions: namely we show that if a geodesically complete, packed, CAT$(0)$-space admits a discrete, cocompact group of isometries with sufficiently small systole then it necessarily splits a non-trivial Euclidean factor. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_10763 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Finiteness of CAT(0) group actions Cavallucci, Nicola Sambusetti, Andrea Group Theory Metric Geometry We prove some finiteness results for discrete isometry groups $Γ$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M = Γ\backslash X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $Γ$ is bounded above by a universal constant only depending on the packing constants and the codiameter. The main tool is a splitting theorem for sufficiently collapsed actions: namely we show that if a geodesically complete, packed, CAT$(0)$-space admits a discrete, cocompact group of isometries with sufficiently small systole then it necessarily splits a non-trivial Euclidean factor. |
| title | Finiteness of CAT(0) group actions |
| topic | Group Theory Metric Geometry |
| url | https://arxiv.org/abs/2304.10763 |