Large embedded balls and Heegaard genus in negative curvature

Bachman, David; Cooper, Daryl; White, Matthew E

doi:10.2140/agt.2004.4.31

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Large embedded balls and Heegaard genus in negative curvature

David Bachman, Daryl Cooper and Matthew E White

Algebraic & Geometric Topology 4 (2004) 31–47

DOI: 10.2140/agt.2004.4.31

arXiv: math.GT/0305290

Abstract

We show if $M$ is a closed, connected, orientable, hyperbolic 3-manifold with Heegaard genus $g$ then $g \geq \frac{1}{2} cosh (r)$ where $r$ denotes the radius of any isometrically embedded ball in $M$ . Assuming an unpublished result of Pitts and Rubinstein improves this to $g \geq \frac{1}{2} cosh (r) + \frac{1}{2} .$ We also give an upper bound on the volume in terms of the flip distance of a Heegaard splitting, and describe isoperimetric surfaces in hyperbolic balls.

Keywords

Heegaard splitting, injectivity radius

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 57M27, 57N16

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Publication

Received: 30 May 2003

Revised: 21 August 2003

Accepted: 29 August 2003

Published: 24 January 2004

Authors

David Bachman
Mathematics Department Cal Poly State University San Luis Obispo CA 93407 USA

Daryl Cooper
Mathematics Department University of California Santa Barbara CA 93106 USA

Matthew E White
Mathematics Department Cal Poly State University San Luis Obispo CA 93407 USA

Volume 4, issue 1 (2004)