The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

Abstract

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type G2. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar questions over more general fields seem to come up for the first time. In the Appendix, we link thedoublecosets with Cech cohomology and indicate connections between certain finiteness properties involving double cosets (Condition (T)) and Bass’s finiteness conjecture in K-theory.