Maximal subgroup growth of some metabelian groups

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 $\mathrm{limsup}\frac{\log m_n(G)}{\log n}$, the degree of polynomial growth of $m_n(G)$). A condition is given for when this upper bound is attained. For $G={\mathbb{Z}}^k⋊\mathbb{Z}$, 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.

44 pages