Morse functions to graphs and topological complexity for hyperbolic 3-manifolds

TL;DR

This paper explores the relationship between topological complexity and metric complexity in hyperbolic 3-manifolds, introducing a new measure called Gromov area based on Morse functions to graphs.

Contribution

It generalizes the concept of width for 3-manifolds using arbitrary adjacency graphs and relates it to Gromov area, a new metric complexity measure.

Findings

Gromov area is linearly related to topological width in hyperbolic 3-manifolds.

The generalized handle adjacency graph provides a broader framework for measuring manifold complexity.

The results connect topological and metric approaches to understanding 3-manifold complexity.

Abstract

Scharlemann and Thompson define the width of a 3-manifold M as a notion of complexity based on the topology of M. Their original definition had the property that the adjacency relation on handles gave a linear order on handles, but here we consider a more general definition due to Saito, Scharlemann and Schultens, in which the adjacency relation on handles may give an arbitrary graph. We show that for compact hyperbolic 3-manifolds, this is linearly related to a notion of metric complexity, based on the areas of level sets of Morse functions to graphs, which we call Gromov area.