Counting Finite Index Subgroups

Let F be a finitely generated group. Denote by an(T) (resp.rnn~ where rn is the number of ideals of O of index n. The function Ck($) is called the Dedekind ("-function of k and expresses much of the arithmetic of k. Similarly, we will study Cr(s) = San(F)n" and describe the first steps of an analogous theory for non-commutative groups. The reader might wonder whether an(F) is the right analogue to rn. One might suggest other possibilities, for example looking at the number of normal subgroups of index n. At this point it is unclear which definition would lead to a richer theory. We, however, have limited ourselves in this survey to the counting of all finite index subgroups. Denote by R(T) the intersection of all finite index subgroups of F. Obviously, an(T/R(T)) = an(T). So, there is no harm in assuming R(T) = {1}, i.e., F is a residually finite group. In this respect, the subgroup growth is