Heegaard genus and Property 'tau' for hyperbolic 3-manifolds

TL;DR

This paper demonstrates that hyperbolic 3-manifolds possess co-final families of finite index subgroups with positive Heegaard gradient, linking geometric properties to algebraic group characteristics.

Contribution

It establishes the existence of co-final subgroup families with Property τ and positive Heegaard gradient for hyperbolic 3-manifolds, connecting group theory and 3-manifold topology.

Findings

Existence of co-final families with Property τ for Kleinian groups

Positive infimal Heegaard gradient in hyperbolic 3-manifolds

Link between algebraic properties and geometric topology

Abstract

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $\tau$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positive.