Heegaard diagrams and handlebody groups
type: Item
instance of: scholarly article
title: Heegaard diagrams and handlebody groups
zbMATH Open document ID: 1022.57012
DOI: 10.1016/S0166-8641(02)00142-6
author: Feng Luo
published in: Topology and its Applications
publication date: 2003-05-27T00:00:00Z
review text: A \textit{Heegard diagram on a handlebody} is the isotopy class of a maximal collection of disjoint, pairwise non-isotopic essential simple loops on the boundary surface of the handlebody. In this paper, the author studies Heegard diagrams from a dynamical systems point of view. The dynamical system here is the action of the handlebody group on Thurston's space of measured laminations. The first study of this dynamical system has been carried out by Howard Masur who introduced the notion of a domain of discontinuity for such an action. The author proves the following Theorem : (a) A Heegard diagram is in Masur's domain of discontinuity if and only if it is \textit{strongly irreducible}, that is, if each component of the diagram intersects all meridian discs. (b) In the case of genus at least two, if \(\alpha\) is a strongly irreducible Heegard diagram and if \(d\) is a hyperbolic metric on the boundary of the handlebody, then there is a computable constant \(K>0\) (which depends only on \(\alpha\) and \(d\)) such that \(I(\alpha,\partial D)\geq K l_d(\partial D)\) for all meridian discs \(D\) in the handlebody. Here, \(I(\cdot,\cdot)\) denotes the geometric intersection function and \(l_d(x)\) denotes the length of the geodesic isotopic to a closed curve \(x\). The author notices that part (a) of the theorem follows from earlier work of \textit{E. Starr} [Curves in handlebodies, Thesis, UC Berkeley, 1992] of which he was not aware when he obtained the result.
Mathematics Subject Classification ID: 57N16
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 1917826
zbMATH Keywords: handlebody
zbMATH Keywords: Heegard diagram
zbMATH Keywords: handlebody group
zbMATH Keywords: curve system
zbMATH Keywords: measured laminations space
zbMATH Keywords: Masur domain
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
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full work available at URL: s0166-8641(02)00142-6
OpenAlex ID: W2045802016
description: scientific article; zbMATH DE number 1917826