Maximal subgroup growth of a few polycyclic groups

Abstract

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let \(G_k = \langle x_1, x_2, \dots , x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle\), so \(G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z} \rtimes \dots \rtimes \mathbb{Z}))\). Then for all integers \(k \geq 2\), we calculate \(m_n(G_k)\), the number of maximal subgroups of \(G_k\) of index \(n\), exactly. Also, for infinitely many groups \(H_k\) of the form \(\mathbb{Z}^2 \rtimes G_2\), we calculate \(m_n(H_k)\) exactly.