Injectivity Radius Bounds in Hyperbolic I-Bundle Convex Cores

TL;DR

This paper proves a conjecture related to uniform bounds on the injectivity radius within convex cores of hyperbolic 3-manifolds, specifically for I-bundles over closed surfaces, including cases with cusps.

Contribution

It provides a proof of McMullen's conjecture for hyperbolic I-bundles over closed surfaces, extending understanding of convex core geometry with cusps.

Findings

Establishes a uniform upper bound on injectivity radius in convex cores.

Extends results to include hyperbolic I-bundles with cusps.

Supports the conjecture that convex cores are uniformly congested.

Abstract

A version of a conjecture of McMullen is as follows: Given a hyperbolizable 3-manifold M with incompressible boundary, there exists a uniform constant K such that if N is a hyperbolic 3-manifold homeomorphic to the interior of M, then the injectivity radius based at points in the convex core of N is bounded above by K. This conjecture suggests that convex cores are uniformly congested. We will give a proof in the case when M is an I-bundle over a closed surface, taking into account the possibility of cusps.