Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

TL;DR

This paper establishes lower bounds on the volume of certain hyperbolic 3-manifolds based on their homological properties and introduces new topological results linking Heegaard genus to the Euler characteristic of guts in incompressible surface complements.

Contribution

It provides novel volume bounds for hyperbolic 3-manifolds using homological data and relates Heegaard genus to the topology of incompressible surface complements.

Findings

Hyperbolic 3-manifolds with specific homology have volume > 5.06 or > 3.08.

New topological results connect Heegaard genus to Euler characteristic of guts.

Volume bounds depend on the dimension of H_1(M;Z_2) and properties of the cup product map.

Abstract

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if H_1(M;Z_2) has dimension at least 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 4, and if the image of the cup product map in H^2(M;Z_2) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes (i.e guts) of the complement of an incompressible surface in M.