Heegaard diagrams and handlebody groups
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.
scientific article; zbMATH DE number 1917826