| Abstrakt: |
Let G be a finitely generated group, and 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 lim sup 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. Let G = Z k ⋊ A Z , where a generator of the Z on the right acts (by conjugation) on Z k by multiplication by A ∈ G L (k , 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 C) of the characteristic polynomial of the smallest block. [ABSTRACT FROM AUTHOR] |
