A model for random three--manifolds

TL;DR

This paper investigates the properties of random 3-manifolds formed by gluing tetrahedra, establishing their typical topological and geometric features, including boundary genus, hyperbolic structure, and spectral properties, as the number of tetrahedra grows large.

Contribution

It introduces a probabilistic model for random 3-manifolds and proves almost sure topological and geometric properties, including hyperbolic metrics and spectral gaps, as the number of tetrahedra increases.

Findings

Manifolds are connected with a single boundary component asymptotically.

Heegaard genus grows linearly with the number of tetrahedra.

Manifolds admit a unique hyperbolic metric with totally geodesic boundary asymptotically.

Abstract

We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component. We prove a law of large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number. We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary. We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini--Schramm limit of our sequence of random manifolds.