Computing a Generating Set of Arithmetic Kleinian Groups

Abstract

The goal of this paper is to demonstrate the use of techniques from hyperbolic geometry to compute generating sets of certain subgroups of $S{L}^{+}(2,\mathbb{C})$; specifically, $S{O}^{+}(Q,\mathbb{Z})$ for $Q$ some integral quadratic form of signature $(3,1)$ that does not represent 0. The algorithm is illustrated for the form $Q_7=x_1^2+x_2^2+x_3-7x^4$, and explicit generating matrices are found.