## 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

### Citations:

Zbl 0499.30036; Zbl 0406.53011
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