On Bounded Packing in Polycyclic Groups

TL;DR

This paper proves that subgroups of certain semidirect products of Z^n with Z have bounded packing under specific automorphism conditions, and extends these results to polycyclic groups of length up to 3, introducing coset growth bounds.

Contribution

It establishes bounded packing for subgroups in semidirect products with diagonalizable automorphisms and extends the concept to polycyclic groups, also introducing coset growth bounds.

Findings

Subgroups of semidirect products with diagonalizable automorphisms have bounded packing.

Polycyclic groups of length ≤ 3 have bounded packing.

Bounded coset growth for specific subgroups in semidirect products.

Abstract

In this paper, we show that any subgroup of a semidirect product of Z^n with Z has bounded packing as long as the action of Z on Z^n is by diagonalizable automorphisms all of whose eigenvalues are real. We use this result to show that any subgroup in a polycyclic group of length 3 or less has bounded packing. We also introduce the notion of coset growth and obtain a bound for the coset growth of subgroup H=<t> in the semidirect product of Z^2 with Z.