Four-free groups and hyperbolic geometry

TL;DR

This paper investigates the geometry of hyperbolic 3-manifolds with 4-free fundamental groups, establishing volume lower bounds and topological constraints for manifolds with smaller volume.

Contribution

It provides new geometric bounds for 4-free hyperbolic 3-manifolds and relates volume constraints to the dimension of their first homology groups.

Findings

Volume of such manifolds exceeds 3.44

Manifolds with volume less than 3.44 have first homology dimension at most 7

New geometric and topological bounds for hyperbolic 3-manifolds

Abstract

We give new information about the geometry of closed, orientable hyperbolic 3-manifolds with 4-free fundamental group. As an application we show that such a manifold has volume greater than 3.44. This is in turn used to show that if M is a closed orientable hyperbolic 3-manifold such that vol M < 3.44, then H_1(M;Z/2Z) has dimension at most 7.