Geometry and Rank of Fibered Hyperbolic 3-Manifolds

TL;DR

This paper investigates the geometric properties of fibered hyperbolic 3-manifolds, establishing a relationship between their diameter, injectivity radius, and the rank of their fundamental group.

Contribution

It provides a new result linking the geometric bounds of hyperbolic 3-manifolds to the algebraic rank of their fundamental groups.

Findings

If the manifold has large diameter and bounded injectivity radius, then its fundamental group rank is 2g+1.

The result applies to closed hyperbolic 3-manifolds fibering over the circle.

The paper advances understanding of the interplay between geometry and algebra in hyperbolic 3-manifolds.

Abstract

Assume that M is a closed hyperbolic 3-manifold fibering over the circle with fiber a closed orientable surface of genus g. We show that if M has large diameter and its injectivity radius is bounded below, then the rank of the fundamental group of M is 2g+1.