#### 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
##### 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 $PS{L}_{2}\left(Z\right)$, if there are only finitely many $PS{L}_{2}\left(Z\right)$–equivalence classes of geodesics in $\Gamma$, and if each geodesic in $\Gamma$ is stabilized by an infinite order subgroup of $PS{L}_{2}\left(Z\right)$. For instance, any finite union of closed geodesics on the modular orbifold ${H}^{2}∕PS{L}_{2}\left(Z\right)$ 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 $\Gamma$. (In particular $p\sim p$.) We will see in §3.2 that $\sim$ is an equivalence relation. We let ${Q}_{\Gamma }={S}^{1}∕\sim$ 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
##### Mathematical Subject Classification 2000
Primary: 57S30
Secondary: 54E99, 51M15