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Combinatorics of simple closed curves on the twice punctured torus



In the standard enumeration of homotopy classes of curves on a surface as words in a generating set for the fundamental group it is st very hard problem to discern those that are simple. In this paper we describe how the complex of simple closed curves on a twice punctured torus Sigma may be given a strikingly simple description by representing them as homotopy classes of paths in a groupoid with two base points. Our starting point are the pi(1)-train tracks developed by Birman and Series. These are weighted train tracks parameterizing the simple closed curves on Sigma similar to Thurston's, but they are defined relative to a fixed presentation of pi(1)(Sigma). We approach the problem by cutting the surface into two disjoint "cylinders"; this decomposes the pi(1)-train tracks into two disjoint parts, relative to which all patterns and relations become much more transparent, each part reducing essentially to the well-known case of a once punctured torus. We obtain global coordinates, called pi(1,2)-weights, for simple closed loops. These coordinates can be easily identified with Thurston's projective measured lamination space S-3. We also solve the problem which originally motivated this work by proving a simple relationship between the leading terms of traces of simple loops in a holomorphic family of representations rho: pi(1)(Sigma) --> PSL(2, C) (corresponding to the Maskit embedding of the twice punctured torus) and the pi(1,2)-weights

Topics: QA
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