Maximal subgroup growth of a few polycyclic groups

TL;DR

This paper precisely calculates the maximal subgroup growth functions for specific classes of polycyclic groups, including certain iterated semidirect products of integers, providing exact formulas for the number of maximal subgroups of given index.

Contribution

It provides exact formulas for the maximal subgroup growth of two classes of polycyclic groups, including iterated semidirect products of integers and certain extensions of ^2 by these groups.

Findings

Exact formulas for m_n(G_k) for all k 

Exact formulas for m_n(H_k) for infinitely many groups H_k

Enhanced understanding of subgroup growth in polycyclic groups

Abstract

We give here the exact maximal subgroup growth of two classes of polycyclic groups. Let $G_k = \langle x_1, x_2, ..., x_k \mid x_ix_jx_i^{-1}x_j \text{ for all } i < j \rangle$. So $G_k = \mathbb{Z} \rtimes (\mathbb{Z} \rtimes (\mathbb{Z} \rtimes ... \rtimes \mathbb{Z})$. Then for all $k \geq 2$, 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.