Uniform models and short curves for random 3-manifolds
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| Main Authors: | , , |
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| Format: | Preprint |
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2019
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| _version_ | 1866911056009363456 |
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| author | Feller, Peter Sisto, Alessandro Viaggi, Gabriele |
| author_facet | Feller, Peter Sisto, Alessandro Viaggi, Gabriele |
| contents | We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1.
These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk.
The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the Geometrization Conjecture by Perelman. In particular, we give a proof of Maher's result that random 3-manifolds are hyperbolic that bypasses Geometrization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_09486 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Uniform models and short curves for random 3-manifolds Feller, Peter Sisto, Alessandro Viaggi, Gabriele Geometric Topology Differential Geometry 58C40, 30F60, 20P05 We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk. The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the Geometrization Conjecture by Perelman. In particular, we give a proof of Maher's result that random 3-manifolds are hyperbolic that bypasses Geometrization. |
| title | Uniform models and short curves for random 3-manifolds |
| topic | Geometric Topology Differential Geometry 58C40, 30F60, 20P05 |
| url | https://arxiv.org/abs/1910.09486 |