Celebratio Mathematica — Birman

An interview with Joan Birman about her mathematics

Works connected to Caroline Mary Series

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J. S. Birman and C. Series: “Geodesics with bounded intersection number on surfaces are sparsely distributed,” Topology 24 : 2 (1985), pp. 217–225. MR 793185 Zbl 0568.57006 article

Let \( M \) be a surface of negative Euler characteristic, possibly with boundary, which is either compact or obtained from a compact surface by removing a finite set of points. Let \( D \) be the Poincaré disc. Choose any representation of \( M \) as \( U/\Gamma \), where \( U \subseteq D \) is the universal covering space of \( M \) and \( \Gamma \subset \operatorname{Isom}(D) \). Then the Poincaré metric on \( D \) induces a metric of constant negative curvature on \( M \) and geodesics in \( U \) project to geodesics on \( M \). A geodesic on \( M \) is said to be complete if it is either closed and smooth, or open and of infinite length in both directions. complete geodesics coincide with those which never intersect \( \partial M \). Note that if \( M \) is obtained from a compact surface by removing a finite number of points to form cusps then a complete open geodesic on \( M \) might tend toward infinity along a cusp. In this paper we study the family \( G_k \) of complete geodesics which have at most \( k \) transversal self-intersections, \( k\geq 0 \).