#### 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.

Notes on file formats
Richard Evan Schwartz

Department of Mathematics, University of Maryland

College Park, MD 20742, USA

Email: res@math.umd.edu

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