Geometry of random 3-manifolds

Abstract

We study random 3-manifolds, as introduced by Dunfield and Thurston, from a geometric point of view. Within this framework, work of Maher allows us to equip a typical random 3-manifold with a canonical geometric structure, namely, a hyperbolic metric. By Mostow rigidity, such metric is unique up to isometries and, hence, we can attach to a random 3-manifold geometric invariants such as volume, Laplace and length spectra, diameter. Our goal is to develop tools to compute these invariants and, in general, to get an effective and explicit description of the hyperbolic structure. More precisely, in this thesis we obtain the following results: We compute the coarse growth rate of volume, diameter and spectral gap for a typical family of random 3-manifolds. We show that the volumes of random 3-manifolds obey to a law of large numbers. We find an explicit model manifold that captures, up to uniform bilipschitz distortion, the geometry of a random 3-manifold.