Finite approximation of free groups II: the Theorems of Ash, Herwig-Lascar and Ribes-Zalesskii -- revisited and strengthened

TL;DR

This work unifies Ash's, HL's, and RZ's theorems by embedding their arguments in the theory of inverse monoids, Stallings/Schreier graphs, and relational-structure extensions. It first establishes a full chain of implications HL⇒RZ⇒Ash⇒HL, then introduces a strengthening based on the third author’s group-construction, yielding a finite $A$-generated group $H$ with enhanced commuting-relabelling properties across all finite graphs of size at most $n$. The strengthened results, together with AB0-type expansions, produce sharper model-theoretic and group-theoretic extensions (EPPA-like) and reveal deeper connections between profinite topology, automorphism-extension problems, and finite representations of free groups. Overall, the paper deepens the correspondence among these foundational theorems and provides concrete, finite-constructive witnesses that strengthen classical extension properties.

Abstract

Relations and interactions between the theorems of Ash, Herwig-Lascar and Ribes-Zalesskii are discussed and it is shown that these three theorems are equivalent in the sense that each of them can be derived from each other one. Some strengthening of these theorems that can be obtained by use of the groups provided by the third author's construction are also considered.

Figures (11)

• Figure 1: The structure $\mathfrak{S}$
• Figure 2: The trees $\mathcal{X}, \mathcal{Y}$ and $\mathcal{Z}$ and the cycle graph $\mathcal{C}$ (dotted)
• Figure 3: Well-definedness of the mapping \ref{['eq:desired mapping']}
• Figure 4: $2$-cyclic and $5$-cyclic configurations
• Figure 5: The amalgamation chain $\mathcal{C}$

Theorems & Definitions (85)

• Lemma 2.1: Theorem 3.2 in stephen
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Remark 2.7
• Proposition 2.8: KapMas, Theorem 5.2