Random methods in 3-manifold theory

TL;DR

This paper demonstrates that using random walks on the mapping class group, infinitely many closed hyperbolic 3-manifolds with specified invariants can be constructed, expanding understanding of 3-manifold diversity.

Contribution

It introduces a novel probabilistic approach to constructing 3-manifolds with prescribed properties, linking random methods to topological and geometric invariants.

Findings

Existence of infinitely many hyperbolic 3-manifolds with given Casson invariant and Heegaard genus

Use of random walks on the mapping class group to generate 3-manifolds

Establishment of a probabilistic model for 3-manifold construction

Abstract

We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random walks on the mapping class group of a closed orientable surface.