Modular circle quotients and PL limit sets

Schwartz, Richard Evan

doi:10.2140/gt.2004.8.1

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Modular circle quotients and PL limit sets

Richard Evan Schwartz

Geometry & Topology 8 (2004) 1–34

DOI: 10.2140/gt.2004.8.1

arXiv: math.GT/0401311

Abstract

We say that a collection $Γ$ of geodesics in the hyperbolic plane $H^{2}$ is a modular pattern if $Γ$ is invariant under the modular group $P S L_{2} (Z)$ , if there are only finitely many $P S L_{2} (Z)$ –equivalence classes of geodesics in $Γ$ , and if each geodesic in $Γ$ is stabilized by an infinite order subgroup of $P S L_{2} (Z)$ . For instance, any finite union of closed geodesics on the modular orbifold $H^{2} ∕ P S L_{2} (Z)$ 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 $p \sim q$ if $p$ and $q$ are the endpoints of a geodesic in $Γ$ . (In particular $p \sim p$ .) We will see in §3.2 that $\sim$ is an equivalence relation. We let $Q_{Γ} = S^{1} ∕ \sim$ be the quotient space. We call $Q_{Γ}$ 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

Mathematical Subject Classification 2000

Primary: 57S30

Secondary: 54E99, 51M15

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Publication

Received: 4 February 2003

Accepted: 13 January 2004

Published: 18 January 2004

Proposed: David Gabai

Seconded: Martin Bridson, Walter Neumann

Authors

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

Volume 8, issue 1 (2004)