Hyperbolicity of Orders of Quaternions Algebras and Group Rings

Details Hyperbolicity of Orders of Quaternions Algebras and Group Rings Hyperbolicity of Orders of Quaternions Algebras and Group Rings

2006-10-23

arxiv.org/abs/math/0610699v2

Mathematics

Rings and Algebras

This report is part of the first chapter of the second author's PhD. Thesis. It was included a result in main theorem and a sk

Scientific paper

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.

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