Examples of Fuchsian groups
In this chapter, we study concrete examples of Fuchsian groups and illustrate the results of the previous chapter.
The first family of groups that we will consider consists of geometrically finite free groups, called Schottky groups. Its construction is based on the dynamics of isometries.
The second family comes from number theory. It consists of three non-uniform lattices: the modular group PSL(2,ℤ), its congruence modulo 2 subgroup and its commutator subgroup.
description of a fundamental domain;
shape of the associated topological surface;
properties of its isometries;
study of its limit set;
characterization of its parabolic points.
We will also construct a coding of the limit sets of Schottky groups and of the modular group. We will use this coding in Chap. IV to study the dynamics of the geodesic flow, and in Chap. VII to translate the behavior of geodesic rays on the modular surface into the terms of Diophantine approximations.
KeywordsFundamental Domain Modular Group Fuchsian Group Commutator Subgroup Continue Fraction Expansion
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