Geometry & Topology, Vol. 8 (2004) Paper no. 1, pages 1--34.

Modular circle quotients and PL limit sets

Richard Evan Schwartz

Abstract. We say that a collection Gamma of geodesics in the hyperbolic plane H^2 is a modular pattern if Gamma is invariant under the modular group PSL_2(Z), if there are only finitely many PSL_2(Z)-equivalence classes of geodesics in Gamma, and if each geodesic in Gamma is stabilized by an infinite order subgroup of PSL_2(Z). For instance, any finite union of closed geodesics on the modular orbifold H^2/PSL_2(Z) lifts to a modular pattern. Let S^1 be the ideal boundary of H^2. Given two points p,q in S^1 we write p~q if p and q are the endpoints of a geodesic in Gamma. (In particular p~p.) We show that ~ is an equivalence relation. We let Q_Gamma=S^1/~ be the quotient space. We call Q_Gamma a modular circle quotient. In this paper we will give a sense of what modular circle quotients `look like' by realizing them as limit sets of piecewise-linear group actions

Keywords. Modular group, geodesic patterns, limit sets, representations

AMS subject classification. Primary: 57S30. Secondary: 54E99, 51M15.

DOI: 10.2140/gt.2004.8.1

E-print: arXiv:math.GT/0401311

Submitted to GT on 4 February 2003. Paper accepted 13 January 2004. Paper published 18 January 2004.

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Richard Evan Schwartz
Department of Mathematics, University of Maryland
College Park, MD 20742, USA

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