• Hyperbolic Unit Groups and Quaternion Algebras

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/119/01/0009-0022

• # Keywords

Hyperbolic groups; quaternion algebras; free groups; group rings; units.

• # Abstract

We classify the quadratic extensions $K=\mathbb{Q}[\sqrt{d}]$ and the finite groups 𝐺 for which the group ring $\mathfrak{o}_K[G]$ of 𝐺 over the ring $\mathfrak{o}_K$ of integers of 𝐾 has the property that the group $\mathcal{U}_1(\mathfrak{o}_K[G])$ of units of augmentation 1 is hyperbolic. We also construct units in the $\mathbb{Z}$-order $\mathcal{H}(\mathfrak{o}_K)$ of the quaternion algebra $\mathcal{H}(K)=\left\frac{-1,-1}{k}(\right)$, when it is a division algebra.

• # Author Affiliations

1. Instituto de Matemática e Estatística, Universidade de São Paulo (IME-USP), Caixa Postal 66281, São Paulo, CEP 05315-970, Brasil
2. Centre for Advanced Study in Mathematics, Panjab University, Chandigarh 160 014, India
3. Escola de Artes, Ciências e Humanidades, Universidade de São Paulo (EACH-USP), Rua Arlindo Béttio, 1000, Ermelindo Matarazzo, São Paulo, CEP 03828-000, Brasil

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019