The small index property for free nilpotent groups

TL;DR

This paper proves that infinitely generated free nilpotent and abelian groups possess the small index property, meaning subgroups of small index contain stabilizers of small subsets, with various applications discussed.

Contribution

It establishes the small index property for a broad class of infinitely generated free nilpotent and abelian groups, extending previous results.

Findings

Infinitely generated free nilpotent groups have the small index property.

Infinitely generated free abelian groups have the small index property.

Applications of the property are discussed in the paper.

Abstract

Let F be a relatively free algebra of infinite rank. We say that F has the SMALL INDEX PROPERTY if any subgroup of Gamma=Aut(F) of index at most rank(F) contains the pointwise stabilizer Gamma_(U) of a subset U of F of cardinality less than rank(F). We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.