Computing a Generating Set of Arithmetic Kleinian Groups

TL;DR

This paper develops a hyperbolic geometry-based algorithm to compute generating sets for specific arithmetic Kleinian groups, demonstrated on a quadratic form, with explicit matrices provided.

Contribution

It introduces a novel method combining hyperbolic geometry techniques to compute generators of certain arithmetic Kleinian groups.

Findings

Successfully computed generating sets for the group associated with Q_7

Explicit matrices for the generators are provided

Demonstrates the effectiveness of geometric methods in group computation

Abstract

The goal of this paper is to demonstrate the use of techniques from hyperbolic geometry to compute generating sets of certain subgroups of $SL^+(2,\mathbb{C})$; specifically, $SO^+(Q,\mathbb{Z})$ 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.