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Titel | The Curve Graph and Surface Construction in S x R |
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Autor | Ingrid Irmer |

Publikationsform | Dissertation |

Abstract | Suppose S is an oriented, compact surface with genus at least two. This thesis investigates the \homology curve complex" of S; a modication of the curve complex rst 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 S x R 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 S x R. The intersection number of curves in S x R is dened by
projecting curves into S. It is proven that the best possible bound on the
distance between two curves c_{0} and c_{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 c_{0} and c_{1} to annuli, and partly due to the
small amount of ambiguity in dening 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 c_{0} and c_{1} to annuli. |

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