Commensurable Arithmetic Groups and Volumes

Abstract

In this chapter, we return to considering arithmetic Kleinian and Fuchsian groups and the related quaternion algebras. Recall that the wide commensurability classes of arithmetic Kleinian groups are in one-to-one correspondence with the isomorphism classes of quaternion algebras over a number field with one complex place which are ramified at all the real places. There is a similar one-to-one correspondence for arithmetic Fuchsian groups. Thus, for a suitable quaternion algebra A, let C(A) denote the (narrow) commensurability class of associated arithmetic Kleinian or Fuchsian groups. In this chapter, we investigate how the elements of C(A)are distributed and, in particular, determine the maximal elements of C(A) of which there are infinitely many. Since these groups are all of finite covolume, their volumes are, of course, commensurable. As a starting point to determining these volumes, a formula for the groups Pρ(O 1), where O is a maximal order in A, is obtained in terms of the number-theoretic data defining the number field and the quaternion algebra. This relies critically on the fact that the Tamagawa number of the quotient A A 1 /A k 1 of the idele group A A 1, is 1, as discussed in Chapter 7. From this formula, one can determine the covolumes of the maximal elements of C(A)and show that all of these volumes are integral multiples of a single number. Much of this chapter is based on work of Borel.