On the automorphism groups of hyperbolic manifolds
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866913781676769280 |
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| author | Budney, Ryan Gabai, David |
| author_facet | Budney, Ryan Gabai, David |
| contents | Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the n-manifold N, pointwise. We show that the (n-4)-th homotopy group of Homeo(S^1 \times D^{n-1}) is not finitely-generated for n >= 4 and in particular the topological mapping-class group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (when n >= 4) do not have the homotopy-type of finite CW-complexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic n-manifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n-4)-th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then π_0\Diff_0(N) is infinitely generated with similar results holding topologically. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_05010 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the automorphism groups of hyperbolic manifolds Budney, Ryan Gabai, David Geometric Topology 57M99, 57R52, 57R50, 57N50 Let Diff(N) and Homeo(N) denote the smooth and topological group of automorphisms respectively that fix the boundary of the n-manifold N, pointwise. We show that the (n-4)-th homotopy group of Homeo(S^1 \times D^{n-1}) is not finitely-generated for n >= 4 and in particular the topological mapping-class group of S^1\times D^3 is infinitely generated. We apply this to show that the smooth and topological automorphism groups of finite-volume hyperbolic n-manifolds (when n >= 4) do not have the homotopy-type of finite CW-complexes, results previously known for n >= 11 by Farrell and Jones. In particular, we show that if N is a closed hyperbolic n-manifold, and if Diff_0(N) represents the subgroup of diffeomorphisms that are homotopic to the identity, then the (n-4)-th homotopy group of Diff_0(N) is infinitely generated and hence if n=4, then π_0\Diff_0(N) is infinitely generated with similar results holding topologically. |
| title | On the automorphism groups of hyperbolic manifolds |
| topic | Geometric Topology 57M99, 57R52, 57R50, 57N50 |
| url | https://arxiv.org/abs/2303.05010 |