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Proceedings - Mathematical Sciences

, Volume 119, Issue 1, pp 9–22 | Cite as

Hyperbolic unit groups and quaternion algebras

  • S. O. JuriaansEmail author
  • I. B. S. Passi
  • A. C. Souza Filho
Article

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 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.

Keywords

Hyperbolic groups quaternion algebras free groups group rings units 

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Copyright information

© Indian Academy of Sciences 2009

Authors and Affiliations

  • S. O. Juriaans
    • 1
    Email author
  • I. B. S. Passi
    • 2
  • A. C. Souza Filho
    • 3
  1. 1.Instituto de Matemática e EstatísticaUniversidade de São Paulo (IME-USP)São PauloBrasil
  2. 2.Centre for Advanced Study in MathematicsPanjab UniversityChandigarhIndia
  3. 3.Escola de Artes, Ciências e HumanidadesUniversidade de São Paulo (EACH-USP)São PauloBrasil

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