Maximal subgroup growth of some metabelian groups

TL;DR

This paper investigates the growth rate of the number of maximal subgroups of a given index in certain metabelian groups, providing bounds and exact growth formulas based on algebraic properties.

Contribution

It establishes an upper bound for the maximal subgroup growth degree in polycyclic metabelian groups and characterizes when this bound is achieved, with explicit formulas for specific cases.

Findings

Upper bound for growth degree in polycyclic metabelian groups

Exact polynomial growth rate for groups of the form Z^k ⋊ Z

Growth rate linked to the rational canonical form of automorphisms

Abstract

Let $m_n(G)$ denote the number of maximal subgroups of $G$ of index $n$. An upper bound is given for the degree of maximal subgroup growth of all polycyclic metabelian groups $G$ (i.e., for $\limsup \frac{\log m_n(G)}{\log n}$, the degree of polynomial growth of $m_n(G)$). A condition is given for when this upper bound is attained. For $G = \mathbb{Z}^k \rtimes \mathbb{Z}$, where $A \in GL(k,\mathbb{Z})$, it is shown that $m_n(G)$ grows like a polynomial of degree equal to the number of blocks in the rational canonical form of $A$. The leading term of this polynomial is the number of distinct roots (in $\mathbb{C}$) of the characteristic polynomial of the smallest block.