Maximal subgroup growth of some metabelian groups

Let G be a finitely generated group, and let mn(G) denote the number of maximal subgroups index n. An upper bound is given for degree subgroup growth all polycyclic metabelian groups (i.e., limsup log n, polynomial mn(G)). A condition when this attained. G=Zk⋊AZ, where generator Z on right acts (by conjugation) Zk by multiplication A∈GL(k,Z). It shown that grows like equal to blocks in rational canonical form A. The leading term distinct roots (in C) characteristic smallest block.