On the logical strength of the automorphism groups of free nilpotent groups

TL;DR

This paper demonstrates that the automorphism group of an infinitely generated free nilpotent group can interpret the entire second-order theory of its rank, revealing deep logical properties of these groups.

Contribution

It establishes a first-order interpretability of the second-order theory of the rank within the automorphism group of free nilpotent groups, addressing a problem posed by Shelah.

Findings

Automorphism groups interpret the second-order theory of the group's rank

Results apply to infinitely generated free nilpotent groups

Addresses a problem posed by Shelah regarding logical strength

Abstract

Considering a particular case of a problem posed by Saharon Shelah, we prove that the automorphism group of an infinitely generated free nilpotent group N first-order interprets the full second-order theory of the set rank(N) in the empty language.