Communications in Analysis and Geometry

Volume 30 (2022)

Number 4

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

Pages: 843 – 868



Diane Hoffoss (Department of Mathematics and Computer Science, University of San Diego, California, U.S.A.)

Joseph Maher (CUNY College of Staten Island and CUNY Graduate Center, Staten Island, New York, U.S.A.)


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 closed 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.

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Received 11 September 2017

Accepted 26 September 2019

Published 30 January 2023