Volume 8, issue 1 (2004)

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Modular circle quotients and PL limit sets

Richard Evan Schwartz

Geometry & Topology 8 (2004) 1–34

arXiv: math.GT/0401311


We say that a collection Γ of geodesics in the hyperbolic plane H2 is a modular pattern if Γ is invariant under the modular group PSL2(Z), if there are only finitely many PSL2(Z)–equivalence classes of geodesics in Γ, and if each geodesic in Γ is stabilized by an infinite order subgroup of PSL2(Z). For instance, any finite union of closed geodesics on the modular orbifold H2PSL2(Z) lifts to a modular pattern. Let S1 be the ideal boundary of H2. Given two points p,q S1 we write p q if p and q are the endpoints of a geodesic in Γ. (In particular p p.) We will see in §3.2 that is an equivalence relation. We let QΓ = S1 be the quotient space. We call QΓ 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.

modular group, geodesic patterns, limit sets, representations
Mathematical Subject Classification 2000
Primary: 57S30
Secondary: 54E99, 51M15
Forward citations
Received: 4 February 2003
Accepted: 13 January 2004
Published: 18 January 2004
Proposed: David Gabai
Seconded: Martin Bridson, Walter Neumann
Richard Evan Schwartz
Department of Mathematics
University of Maryland
College Park
MD 20742