Arithmetic rigidity and units in group rings (Q2731959)

Two groups \(\Gamma_1\) and \(\Gamma_2\) which have a common subgroup of finite index are called `commensurable'. The main result of the paper under review is to prove that if the unit groups of the integral group rings of two finite groups are commensurable then big factors of the group algebras over the rational numbers are isomorphic. The small factors of the group algebra over the rationals are commutative pieces, quaternion algebras and negative definite quaternion algebras over a totally real field. If, except in the trivial way, none of these small pieces occurs, then the converse of the result is true as well.NEWLINENEWLINENEWLINEThe author's method used for the proof here is very different from the methods used in earlier papers by del Rio, Jespers and others on the subject. In the paper under review the author uses a beautiful mixture of the theory of algebraic and arithmetic groups as well as algebraic number theory. As an example of this variety of arguments we mention that fundamental groups of orientable surfaces of genus bigger than 2 play a role in the proof, as well as Galois cohomology.