Hyperbolicity of Orders of Quaternions Algebras and Group Rings

2006/10/23 by Filho, S. O. Juriaans A. C. Souza
#16S34 #20C99 #FOS: Mathematics #Group Theory (math.GR) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.math/0610699

Abstract

For a given divison algebra of the quaternions we construct two types of units: Pell units and Gauss units. If K is a rational quadratic extension and G is a finite group, we classify R and G, s.t., the unit group U(RG) of augmentation one is hyperbolic. In particular we give necessary and sufficient conditions when G is the quaternion group of order 8. In this case, the hyperbolic boundary is isomorphic to the 2-dimensional sphere. As a result we reach a class of groups of one end, that are not virtually free.

Related

• Hyperbolic Unit Groups and Quaternion Algebras
• On Units in Orders in 2-by-2 Matrices over Quaternion Algebras with Rational Center
• Free Groups in Quaternion Algebras
• Group rings whose group of units is hyperbolic
• Presentations of Groups Acting Discontinuously on Direct Products of Hyperbolic Spaces