On a classification of limit points of infinitely generated Schottky groups
type: Item
instance of: scholarly article
title: On a classification of limit points of infinitely generated Schottky groups
zbMATH Open document ID: 0963.30028
DOI: 10.1023/A:1009556612040
author: A. N. Starkov
author: Françoise Dal'Bo
published in: Journal of Dynamical and Control Systems
publication date: 2001-06-22T00:00:00Z
review text: Consider the Poincaré model of the hyperbolic plane, i.e. the unit disc \(\Delta\) equipped with the metric \(dl=2|dz|/(1-|z|^2)\). Geodesic lines are the arcs of circles orthogonal to \(S=\partial \Delta\) at both ends (including diameters); horocycles are the circles tangent to \(S\). The group \(G\simeq \text{PSL} (2,{\mathbb R})\) of all orientation-preserving isometries of \(\Delta\) acts freely and transitively on \(T^1\Delta\). A discrete subgroup \(\Gamma \subset G\) is called a Fuchsian group. One considers also the Lobachevsky model (i.e. the upper half-plane \({\mathbb H}^2=\{ y>0\}\) equipped with the metric \(dl=|dz|/\operatorname {Im}z)\), the geodesics (horocycles) being the half-circles or half-lines orthogonal to \(\{ y=0\}\) (resp. the circles tangent to it). A classical Schottky group is a Fuchsian group with generators \(s_1, \dots,s_n\) obtained by placing \(2n\) disjoint closed semidiscs \(B_1,\dots, B_n, B_{-1},\dots,B_{-n}\) centered on \(\{ y=0\}\) and pairing them in such a way that \(s_i(\text{Int} (B_i))= \text{Ext}(B_{-i})\). Given a reference point \(w\), introduce the limit set \(\Lambda \subset S\) as the set of accumulation points for the orbit \(\Gamma (w)\). The behaviour of geodesic and horocycle trajectories on \(\Gamma \backslash T^1{\mathbb H}^2\) allows one to introduce subclasses of \(\Lambda\) such as the conical, horocyclic etc. limit sets \(\Lambda _c\), \(\Lambda _h\). For a class of infinitely generated Schottky groups (obtained by placing countably many disjoint semicircles in \({\mathbb H}^2\) with a single accumulation point in \(\partial {\mathbb H}^2\), and pairing them by hyperbolic transformations) a complete description of \(\Lambda _c\) and of other subclasses of \(\Lambda\) is given in terms of natural combinatorial coding of limit points; this coding is the same for all Fuchsian groups. In contrast, the structure of horocycle orbits is sensitive to geometric parameters of the semicircles near the accumulation …
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 1545853
zbMATH Keywords: Poincaré and Lobachevsky model of the hyperbolic plane
zbMATH Keywords: Fuchsian group
zbMATH Keywords: Schottky group
zbMATH Keywords: geodesic
zbMATH Keywords: horocycle
zbMATH Keywords: Garnett limit points
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1545853