Uniform models and short curves for random 3-manifolds.

We give two constructions of hyperbolic metrics on Heegaard splittings satisfying certain conditions that only use tools from the deformation theory of Kleinian groups. In particular, we do not rely on the solution of the Geometrization Conjecture by Perelman. Both constructions apply to random Heegaard splitting with asymptotic probability 1. The first construction provides explicit uniform bilipschitz models for the hyperbolic metric. The second one gives a general criterion for a curve on a Heegaard surface to be a short geodesic for the hyperbolic structure, such curves are abundant in a random setting. As an application of the model metrics, we discuss the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and questions about arithmeticity and commensurability in families of random 3-manifolds.