Volume and topology of bounded and closed hyperbolic 3-manifolds

TL;DR

This paper establishes volume lower bounds for certain hyperbolic 3-manifolds based on topological properties like genus and homology, using a dichotomy involving return paths and embedded submanifolds.

Contribution

It introduces new volume bounds for hyperbolic 3-manifolds with specified boundary and homological conditions, employing a novel dichotomy approach.

Findings

Manifolds with genus ≥ 5 have volume > 6.89

Manifolds with certain homology conditions have volume > 3.44

Dichotomy involving return paths and embedded submanifolds

Abstract

Let N be a compact, orientable hyperbolic 3-manifold with connected, totally geodesic boundary of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 6.89. The proof of this result uses the following dichotomy: either N has a long return path (defined by Kojima-Miyamoto), or N has an embedded, codimension-0 submanifold X with incompressible boundary $T \sqcup \partial N$, where T is the frontier of X in N, which is not a book of I-bundles. As an application of this result, we show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 5, and if the image in H^2(M;Z_2) of the cup product map has image of dimension at most 1, then M has volume greater than 3.44.