Subgroup growth of free products
For a group \(G\) denote by \(s_G(n)\) the number of subgroups of index \(n\) in \(G\). The purpose of the present paper is to investigate the growth behaviour and the asymptotics of \(s_G(n)\) in the case when \(G\) is a free product of the form \(G=*^s_{k=1}G_k*F_r\) (\(0\leq r,s<\infty\), \(1<m_k=|G_k|<\infty\)). Here, the \(G_k\) are arbitrary finite groups and \(F_r\) denotes the free group of rank \(r\). The main result gives a complete asymptotic expansion for \(s_G(n)\) in terms of the numbers \(r\), \(s\), \(m_k\), \(s_{G_k}(d_k)\), \(d_k|m_k\). The author also considers the question as to what kind of information on this decomposition of a group \(G\) is contained in the function \(s_G(n)\) or its asymptotics, and he shows that this combinatorial information determines \(G\) up to finitely many isomorphism types.
scientific article; zbMATH DE number 931986