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    • 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


      S O Juriaans1 I B S Passi2 A C Souza Filho3

      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
    • Dates

  • Proceedings – Mathematical Sciences | News

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      Posted on July 25, 2019

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