The Modular Group and Its Subgroups

  • Xueli Wang
  • Dingyi Pei

Abstract

Let
$$ SL_2 (\mathbb{R}) = \left\{ {\left. {\left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right)} \right|a,b,c,d \in \mathbb{R},ad - bc = 1} \right\}. $$
For any \( \sigma = \left( {\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} } \right) \in SL_2 (\mathbb{R}) \) define a transformation on the whole complex plane as follows
$$ \sigma (z) = \frac{{az + b}} {{cz + d}}. $$
It is easy to prove
$$ \operatorname{Im} \left( {\sigma (z)} \right) = \frac{{\operatorname{Im} \left( z \right)}} {{\left| {cz + d} \right|^2 }}. $$
.

Keywords

Equivalence Class Conjugate Classis Discrete Subgroup Modular Group Fuchsian Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Science Press Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Xueli Wang
    • 1
  • Dingyi Pei
    • 2
  1. 1.Department of MathematicsSouth China Normal UniversityGuangzhouChina
  2. 2.Institute of Mathematics and Information ScienceGuangzhou UniversityGuangzhouChina

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