The Curve Graph and Surface Construction in <em>S x R</em>

Abstract

Suppose <em>S</em> is an oriented, compact surface with genus at least two. This thesis investigates the \homology curve complex" of <em>S</em>; a modification of the curve complex first studied by Harvey in which the verticies are required to be homologous multicurves. The relationship between arcs in the homology curve graph and surfaces with boundary in <em>S x R</em> is used to devise an algorithm for constructing efficient arcs in the homology multicurve graph. Alternatively, these arcs can be used to study oriented surfaces with boundary in <em>S x R</em>. The intersection number of curves in <em>S x R</em> is defined by projecting curves into <em>S</em>. It is proven that the best possible bound on the distance between two curves <em>c</em>₀ and <em>c</em>₁ in the homology curve complex depends linearly on their intersection number, in contrast to the logarithmic bound obtained in the curve complex. The difference in these two results is shown to be partly due to the existence of what Masur and Minsky refer to as large subsurface projections of <em>c</em>₀ and <em>c</em>₁ to annuli, and partly due to the small amount of ambiguity in defining this concept. A bound proportional to the square root of the intersection number is proven in the absence of a certain type of large subsurface projection of <em>c</em>₀ and <em>c</em>₁ to annuli.