Volume and homology of one-cusped hyperbolic 3-manifolds

TL;DR

This paper establishes lower bounds on the volume of one-cusped hyperbolic 3-manifolds under specific algebraic and topological conditions related to their homology and fundamental group structure.

Contribution

It provides new volume bounds for hyperbolic 3-manifolds based on homological and subgroup constraints, extending previous geometric-topological results.

Findings

Volume > 5.06 under certain homological conditions

Volume > 5.06 if no genus-2 incompressible surface in the core

Results connect algebraic properties to geometric volume

Abstract

Let M be a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp. If we assume that pi_1(M) has no subgroup isomorphic to a genus-2 surface group, and that either (a) H_1(M;Z_p) has dimension at least 5 for some prime p, or (b) H_1(M;Z_2) has dimension at least 4, and the subspace of H^2(M;Z_2) spanned by the image of the cup product has dimension at most 1, then vol M > 5.06 If we assume that H_1(M;Z_2) has dimension at least 7, and that the compact core of M does not contain a genus-2 closed incompressible surface, then vol M > 5.06.