Thick hyperbolic 3-manifolds with bounded rank

TL;DR

This paper develops a geometric decomposition for thick hyperbolic 3-manifolds with bounded rank, providing bounds on topological and geometric invariants such as Heegaard genus and embedded ball radius.

Contribution

It introduces a new decomposition technique for convex cores of hyperbolic 3-manifolds with bounded rank, leading to explicit bounds on key geometric quantities.

Findings

Upper bounds on Heegaard genus in terms of rank and injectivity radius

Bounds on the radius of embedded balls in the convex core

Construction of a geometric decomposition for the convex core

Abstract

We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.