Hyperbolic Unit Groups and Quaternion Algebras

TL;DR

This paper classifies quadratic extensions and finite groups where the unit group of the group ring over integers is hyperbolic, and constructs units in quaternion algebras, advancing understanding of algebraic structures with hyperbolic properties.

Contribution

It provides a classification of quadratic extensions and finite groups with hyperbolic unit groups in their group rings, and constructs units in quaternion algebras, offering new insights into algebraic structures.

Findings

Classification of quadratic extensions and finite groups with hyperbolic unit groups

Construction of units in non-split quaternion algebras

Identification of conditions for hyperbolicity in algebraic structures

Abstract

We Classify the rational quadratic extensions K and the finite groups G for which the group ring R[G] of G over the ring R of integers of K has the property that the group of units of augmentation 1 of R[G] is hyperbolic. We also construct units in a non-split quaternion algebra over R.