Statistics of isomorphism types in free products

Let Γ be a free product of finitely many finite- and infinite-cyclic groups. For a subgroup Δ of finite index given by its coset representation we compute its isomorphism type, i.e., its decomposition as a free product of finite- and infinite-cyclic groups. We determine the set of isomorphism types realized by finite-index subgroups, the asymptotics of the subgroup numbers with prescribed isomorphism types, and the distribution of the isomorphism types among subgroups of fixed index. Apart from group-theoretic arguments, the proofs of the present paper make use of asymptotic, combinatorial, and probabilistic ideas and techniques.