Classification and Statistics of Finite Index Subgroups in Free Products

The principal theme of this paper is the enumeration of finite index subgroups Delta in a free product Gamma of finite groups under various restrictions on the isomorphism type of Delta. In particular, we completely resolve the realization, asymptotic, and distribution problems for free products Gamma of cyclic groups of prime order (prior to this work, these questions were wide open even in the case of the classical modular group). This complex of problems, usually referred to as Poincare-Klein Problem, originally arose around 1890 out of the work of Klein and Poincare on automorphic functions and related number theory, but has also grown roots in geometric function theory and, more recently, in the theory of subgroup growth. Ideas and techniques from the theory of generalized permutation representations (an enumerative theory of wreath product representations recently developed by the first named author) play a fundamental role here. Other tools come from analytic number theory, combinatorics, and probability theory