Statistics of isomorphism types in free products.
Let \(\Gamma=G_1*G_2*\cdots*G_s*F_r\) be a free product of finite groups \(G_i\) and a free group \(F_r\) of rank \(r\) and let \(U_1,\dots,U_t\) be a complete list of isomorphism types of non-trivial subgroups of \(G_1,\dots,G_s\). Every finite-index subgroup \(\Delta\) of \(\Gamma\) is of the form \(\Delta\cong U_1^{*\lambda_1}*U_2^{*\lambda_2}*\cdots*U_t^{*\lambda_t}*F_\mu\) with non-negative integers \(\lambda_1,\dots,\lambda_t,\mu\). The tuple \((\lambda_1,\dots,\lambda_t,\mu)\) is called the isomorphism type of \(\Delta\). The first aim of the article is an algorithmic determination of the isomorphism type of a subgroup of finite index given by its coset representation. The authors 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 make use of asymptotic, combinatorial, and probabilistic ideas and techniques.
scientific article; zbMATH DE number 5706204