Modular circle quotients and PL limit sets

Schwartz, Richard Evan¹

¹ Department of Mathematics, University of Maryland, College Park, MD 20742, USA

Geometry & topology, Tome 8 (2004) no. 1, pp. 1-34.

Résumé

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.

Détail

DOI : 10.2140/gt.2004.8.1

Keywords: modular group, geodesic patterns, limit sets, representations

Bibliographie

[1] L Bers, On boundaries of Teichmüller spaces and on Kleinian groups I, Ann. of Math. $(2)$ 91 (1970) 570

[2] M Bonk, O Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000) 266

[3] F P Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley Sons (1987)

[4] B Maskit, Kleinian groups, Grundlehren der Mathematischen Wissenschaften 287, Springer (1988)

[5] J G Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer (1994)

[6] C Series, Geometric methods of symbolic coding, from: "Ergodic theory, symbolic dynamics, and hyperbolic spaces" (editors T Bedford, M Keane, C Series), Oxford Science Publications, The Clarendon Press Oxford University Press (1991) 125

[7] R E Schwartz, Degenerating the complex hyperbolic ideal triangle groups, Acta Math. 186 (2001) 105

[8] R E Schwartz, Circle quotients and string art, Topology 41 (2002) 495

[9] R Schwartz, Pappus' theorem and the modular group, Inst. Hautes Études Sci. Publ. Math. (1993)