On Restricting Subsets of Bases in Relatively Free Groups

TL;DR

This paper investigates how subsets of bases in various types of free groups relate to bases of their subgroups, showing that certain subsets generate subgroups with bases contained within smaller generating sets.

Contribution

It establishes conditions under which subsets of bases in free groups are contained in bases of smaller subgroups, extending understanding of basis restrictions in free groups.

Findings

Subsets expressed without certain generators are contained in bases of smaller free subgroups.

Results apply to free groups, free abelian groups, free nilpotent, and free solvable groups.

Most cases confirm the subset's basis is contained within a smaller generating set.

Abstract

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety A_mA_n, and let A = {a_1,..., a_r} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {a_{l+1},...,a_r}, then S is a subset of a basis for the relatively free group on {a_1,...,a_l}.