Conciseness of first-order formulae

TL;DR

The work generalizes conciseness from words to first-order formulae in groups, proving that all formulae are concise in the abelian class (via $G_\varphi$ and $\varphi(G)$) and that every existential formula is concise in torsion-free locally class-$2$ nilpotent groups, using Mal'cev bases and Hall polynomials. It develops the weakly rational framework to produce a wide array of concise formulae in residually finite groups, including constructions that preserve weak rationality under disjoint products and ena formulae for outer commutator words, and it connects these results to residual/strongly residual notions with witness sets. The paper further extends conciseness to topological groups, showing that negative formulae are concise in compact Hausdorff groups, and establishes that openness of certain definable sets yields concise behavior in this setting. Overall, it provides a broad, constructive toolkit for obtaining concise formulae across algebraic and topological group contexts, highlighting both complete results and directions for future work.

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.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Schur