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Simple Closed Curves on the Twice Punctured Torus

Caroline Series
Department of Mathematics, Warwick University


ABSTRACT

Homotopy classes of closed loops on a surface $\Sigma$ are traditionally studied as conjugacy classes in the fundamental group $\pi_1(\Sigma)$; simple loops are parameterised using Thurston's theory of weighted train tracks. These two points of view are brought together by the use of $\pi_1$-train tracks (well known to JSB), a method which relates the weights directly to elements of $\pi_1(\Sigma)$. However the patterns in words which represent simple closed curves are hard to discern and yet harder to use in calculations. In this talk we describe how simple closed curves on a twice punctured torus may be given a strikingly simple description by representing them as homotopy classes of {\em paths} thought of as elements in a ``fundamental groupoid'' with two base points. Cutting the surface into two disjoint ``cylinders'' decomposes the $\pi_1$-train tracks into two disjoint parts, relative to which all patterns and relations become much more transparent, each part being essentially the well known case of a once punctured torus. Besides obtaining global coordinates for simple closed loops, allowing easy identification with Thurston's projective measured lamination space $S^3$, we solve the problem which originally motivated this work by relating traces of simple loops in a certain holomorphic family of representations $\rho: \pi_1(\Sigma) \to PSL(2,\CC)$ to these `` $\pi_1,2$-weights''. The proof involves factoring $\rho$ through a representation of the associated groupoid, after which the use of cutting sequences and $\pi_1,2$-weights makes some very complicated calculations tractably simple.