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Geodesics with bounded intersection number on surfaces are sparsely distributed. (English) Zbl 0568.57006

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 V/\(\Gamma\), where \(V\subset D\) is the universal covering space of M and \(\Gamma\) \(\subset Isom(D)\). Then the Poincaré metric on D induces a metric of constant negative curvature on M and geodesics in V 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. For each \(k\geq 0\), let \(G_ k\) be set of complete geodesics on M which have at most k transversal self- intersections.
By techniques of hyperbolic geometry and parametrization of geodesic arcs suggested by the Dehn-Thurston parametrization for simple closed curves, the authors prove: Theorem 1: For each \(k\geq 0\), the set \(S_ k\) of points of M which lie on a geodesic of \(G_ k\) is nowhere dense and has Hausdorff dimension one. Theorem 2: The set of points of (\(\partial D\times \partial D-diagonal)\) which represent endpoints of geodesics in D whose projection on M is in \(G_ k\), is nowhere dense and has Hausdorff dimension zero. Theorem 3: The set of tangent vectors in the unit tangent bundle \(T_ 1M\) which project onto tangents to geodesics in \(G_ k\) is nowhere dense and has Hausdorff dimension one, with respect to any natural choice of metric in \(T_ 1M\). Theorem 4: If two closed simple geodesics on M have exactly one intersection point, \(x\in M\), then infinitely many closed simple geodesics pass through x.
Theorem 1 answers a question of T. Jorgensen [Proc. Am. Math. Soc. 86, 120-122 (1982; Zbl 0499.30036)] as to whether \(S_ 0\) has measure zero and also Abikoff’s question [op. cit.] as to whether \(S_ 0\) is dense. Theorem 4 is a slight extension of a result of T. Jorgensen [ibid. 72, 140-142 (1978; Zbl 0406.53011)]. Finally the authors establish two interesting open problems about the sets \(S_ k\) and \(G_ k\).
Reviewer: E.Outerelo

MSC:

57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
53C22 Geodesics in global differential geometry
30F10 Compact Riemann surfaces and uniformization
57M10 Covering spaces and low-dimensional topology
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