Hyperbolic volume, Heegaard genus and ranks of groups

TL;DR

This paper explores the relationships between hyperbolic volume, Heegaard genus, and group ranks in 3-manifolds, proposing conjectures that connect these geometric and algebraic properties.

Contribution

It formulates new conjectures linking Heegaard genus and group ranks to hyperbolic volume, suggesting implications for understanding 3-manifold structures.

Findings

Proposed conjectures relating Heegaard genus and group ranks to hyperbolic volume

Derived new implications for the properties of 3-manifolds based on these conjectures

Established connections that could guide future research in 3-manifold topology

Abstract

Some conjectures about Heegaard genera and ranks of fundamental groups of 3-manifolds are formulated, and it is shown that they imply new statements about hyperbolic volume.