Injectivity Radius Bounds in Hyperbolic Convex Cores I

TL;DR

This paper proves a conjecture that hyperbolic 3-manifolds with certain boundary conditions have uniformly bounded injectivity radius in their convex cores, extending previous results to more complex manifold types.

Contribution

The paper establishes the injectivity radius bounds for hyperbolic 3-manifolds that are books of I-bundles or acylindrical, expanding the class of manifolds where the conjecture holds.

Findings

Injectivity radius is bounded in convex cores of specified 3-manifolds.

Bound depends on the number of generators of the fundamental group for books of I-bundles.

Extends previous work to acylindrical hyperbolizable 3-manifolds.

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. In previous work, the author has proven the conjecture for $I$-bundles over a closed surface, taking into account the possibility of cusps. In this paper, we establish the conjecture in the case that M is a book of I-bundles or an acylindrical, hyperbolizable 3-manifold. In particular, we show that if M is a book of I-bundles, then the bound on injectivity radius depends on the number of generators in the fundamental group of M.