Conciseness of first-order formulae

TL;DR

This paper extends the concept of conciseness from words to first-order formulae in group theory, establishing new results for various classes of groups and constructing examples of concise formulae.

Contribution

It proves all formulae are concise in abelian groups and that existential formulae are concise in torsion-free locally class-2 nilpotent groups, introducing new examples of weakly rational words.

Findings

All formulae are concise in abelian groups.

Existential formulae are concise in torsion-free locally class-2 nilpotent groups.

Constructed new weakly rational words for residually finite groups.

Abstract

A word $w$ is concise in a class of groups $\mathcal{C}$ if, for every group $G$ in $\mathcal{C}$, the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in $G$. This notion can be naturally extended to first-order formulae in the language of groups. We consider this more general setting and establish conciseness for various classes of groups and formulae. We prove that all formulae are concise in the class of abelian groups and that every existential formula is concise in the class of torsion-free locally class-2 nilpotent groups. In addition, we construct new examples of weakly rational words, which allow us to produce a wide variety of formulae that are concise in the class of residually finite groups.