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April 10, 2015 By

Hyperbolicity of quadratic fields, semigroup algebras and $RA$-loops

C. R. Math. Rep. Acad. Sci. Canada Vol. 28 (4) 2006, pp. 105–113

December 30, 2006

S.O. Juriaans, Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Caixa Postal 66281, Sao Paulo, CEP 05315-970 Brasil; email: ostanley@ime.usp.br

A.C. Souza Filho, Instituto de Matematica e Estatistica, Universidade de Sao Paulo, Caixa Postal 66281, Sao Paulo, CEP 05315-970 Brasil; email: calixto@ime.usp.br

Abstract/Résumé:

For the rational extension \(K=\Q \sqrt{-d}\) with \(d\) a square free integer and \(R\) the ring of algebraic integers of \(K\), we classify \(R\) and \(G\) such that \(\U_1(RG)\) is a hyperbolic group. In particular, the unit group \(\U_1(RK_8)\) is hyperbolic if and only if \(d>0\) and \(d \equiv 7 \pmod 8\). In this case, the hyperbolic boundary \(\partial(U_1(RK_8))\) is isomorphic to \(S^2\), the two-dimensional sphere. Thus, \(\U_1(RK_8)\) is a hyperbolic group of one end. Also, for a given division algebra of the quaternions, we construct two types of units of its \(\Z\)-orders: Pell’s units and Gauss’ units. Next, we classify the finite semigroups \(S\) such that for all \(\Z\)-orders \(\Gamma\) of the algebra \(\Q S\), the unit group \(\U(\Gamma)\) is hyperbolic. Finally, we classify the \(RA\)-loops \(L\) for which the unit loop of its integral loop ring does not contain a free abelian subgroup of rank two.

Nous classifions les anneaux d’entiers des extensions quadratiques rationelles, que nous noterons \(R\), tel que le groupe d’unités \(\U(RG)\) sur ces anneaux est hyperbolique pour un certain groupe \(G\) fix' e. En particulier, le groupe \(\U_1(RK_8)\) est hyperbolique si et seulement si \(d>0\) et \(d \equiv 7 \pmod 8\). Dans ce cas, la frontière hyperbolique \(\partial(U_1(RK_8))\) est isomorphe à la sphère \(S^2\) de dimension \(2\). Nous considérons une algèbre de quaternions qui est aussi une algèbre de division. Pour un \(\Z\)-ordre de cette algèbre, nous présentons des constructions de deux types d’unités: les unités de Gauss et les unités de Pell. Par la suite, nous classifions les semi-groupes finis \(S\) dont l’algébre unitaire \(\Q S\) verifie la propri' et' e suivante: pour tout \(\Z\)-ordre \(\Gamma\) de \(\Q S\) le groupe d’unit' es \(\U (\Gamma)\) est hyperbolique. Dans le m^ eme contexte, nous classifions les \(RA\)-loops \(L\) dont le loop d’unités ne contient aucun sous-groupe abelien libre de rang \(2\).

Keywords:
AMS Subject Classification: Units; groups of units 16U60

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup cycles of ideals elliptic curves fixed point Fourier transform function fields. general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping noninterlacing numerical range orthogonal polynomials Predual space prime number property SP quadratic forms Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces Weak Markov set Whitney problems

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