Jørgensen Lemma

In this thesis we will first do some classification of the elements in \(PSL(2,\mathbb{R})\). After that we will introduce the notion of Fuchsian groups, i. e. a discrete subgroup of \( PSL( 2, \mathbb{R})\), and prove two criteria for discreteness in \(PSL(2,\mathbb{R})\). One of them is the Jørgensen Inequality, which is the main theorem in this thesis. Finally we look at the special case of equality in the Jørgensen inequality.