Presentations of Groups Acting Discontinuously on Direct Products of Hyperbolic Spaces

TL;DR

This paper explores extending Poincaré's Polyhedron Theorem to describe unit groups of orders in algebraic structures acting on hyperbolic spaces, specifically applying it to the Hilbert modular group.

Contribution

It initiates a novel approach to describe unit groups in algebraic orders via discontinuous actions on hyperbolic spaces, focusing on degree 2 algebra components.

Findings

Constructed a fundamental domain for the Hilbert modular group.

Extended Poincaré's method to a product of hyperbolic spaces.

Provided a framework for presentations of unit groups in specific algebraic contexts.

Abstract

The problem of describing the group of units $\mathcal{U}(\mathbb{Z} G)$ of the integral group ring $\mathbb{Z} G$ of a finite group $G$ has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order $\mathcal{O}$ in the simple epimorphic images $A$ of the rational group algebra $\mathbb{Q} G$. Making use of the presentation part of Poincar\'e's Polyhedron Theorem, Pita, del R\'io and Ruiz proposed such a method for a large family of finite groups $G$ and consequently Jespers, Pita, del R\'io, Ruiz and Zalesskii described the structure of $\mathcal{U}(\mathbb{Z} G)$ for a large family of finite groups $G$. In order to handle many more groups, one would like to extend Poincar\'e's Method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra $A$ has degree 2 then via the Galois embeddings of the centre of the algebra $A$ one considers the group of reduced norm one elements of the order $\O$ as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of $\mathbb{Q} G$ and in particular describe the unit group of $\mathbb{Z} G$ for every group $G$ with irreducible character degrees less than or equal to 2. The aim of this paper is to initiate this approach by executing this method on the Hilbert modular group, i.e. the projective linear group of degree two over the ring of integers in a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two. The fundamental domain constructed is an analogue of the Ford domain of a Fuchsian or a Kleinian group.