Modular circle quotients and PL limit sets

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.