Uniform models and short curves for random 3manifolds
Abstract
We provide two constructions of hyperbolic metrics on 3manifolds 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 3manifolds. 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 3manifolds are hyperbolic that bypasses Geometrization.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 DOI:
 10.48550/arXiv.1910.09486
 arXiv:
 arXiv:1910.09486
 Bibcode:
 2019arXiv191009486F
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Differential Geometry;
 58C40;
 30F60;
 20P05
 EPrint:
 68 pages, 4 figures. Version 3: Rewritten introduction with focus on the applications to injectivity radius, diameter, and nonarithmeticity and noncommesurability of random 3manifolds. Fundamentally reworked the proof concerning the model metrics