On a classification of limit points of infinitely generated Schottky groups
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 point as shown in examples.
Poincaré and Lobachevsky model of the hyperbolic plane
scientific article; zbMATH DE number 1545853