Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds

TL;DR

This survey explores how geometric techniques inform the understanding of topological features like Heegaard genus and fundamental group rank in hyperbolic 3-manifolds, linking combinatorial and geometric perspectives.

Contribution

It provides an overview of how geometric methods are applied to analyze topological properties of hyperbolic 3-manifolds, highlighting recent connections between geometry and topology.

Findings

Geometric methods offer insights into Heegaard genus and fundamental group rank.

Connections between combinatorial descriptions and geometric properties are established.

Survey summarizes recent advances in the field.

Abstract

In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating combinatorial descriptions and geometric properties of hyperbolic 3-manifolds.