Finite approximation of free groups I: the $F$-inverse cover problem

TL;DR

The paper resolves the finite $F$-inverse cover problem by constructing a finite $E$-generated group $G$ that mirrors the Cayley-graph geometry of a given graph and supports a content function, ensuring that path relations can be rewritten using only content edges. The approach combines retracability, coset- and cluster-extensions, and a pair of inductive procedures (forward and upward induction) to build a chain of expansions culminating in $G$ that preserves automorphisms of the input graph. The main technical achievement is a lemma guaranteeing that every finite connected oriented graph admits a symmetry-extended, content-respecting finite $E$-group, which yields a finite $F$-inverse cover for any finite inverse monoid. This offers a symmetry-preserving finite approximation of free-group behavior within finite groups and substantiates the finite $F$-inverse cover for finite inverse monoids, advancing the Henckell–Rhodes program and linking inverse monoid theory with finite-group constructions.

Abstract

For a finite connected graph $\mathcal{E}$ with set of edges $E$, a finite $E$-generated group $G$ is constructed such that the set of relations $p=1$ satisfied by $G$ (with $p$ a word over $E\cup E^{-1}$) is closed under deletion of generators (i.e.~edges). As a consequence, every element $g\in G$ admits a unique minimal set $\mathrm{C}(g)$ of edges (the \emph{content} of $g$) needed to represent $g$ as a word over $\mathrm{C}(g)\cup\mathrm{C}(g)^{-1}$. The crucial property of the group $G$ is that connectivity in the graph $\mathcal{E}$ is encoded in $G$ in the following sense: if a word $p$ forms a path $u\longrightarrow v$ in $\mathcal{E}$ then there exists a $G$-equivalent word $q$ which also forms a path $u\longrightarrow v$ and uses only edges from their content; in particular, the content of the corresponding group element $[p]_G=[q]_G$ spans a connected subgraph of $\mathcal{E}$ containing the vertices $u$ and $v$. As the free group generated by $E$ obviously has these properties, the construction provides another instance of how certain features of free groups can be ``approximated'' or ``simulated'' in finite groups. As an application it is shown that every finite inverse monoid admits a finite $F$-inverse cover. This solves a long-standing problem of Henckell and Rhodes.

Figures (12)

• Figure 1: Coset $2$- and $3$-cycles
• Figure 2: Configuration as in the proof of Lemma \ref{['lem:3-acyclic']}
• Figure 3: Configurations violating admissibility (Definition \ref{['def:admissible']})
• Figure 4: Parts of $\mathcal{K}\cup\bigcup_{t=1}^{k} v_t\mathcal{G}[B]\subseteq\mathcal{G}[A]$ and of $\mathsf{CE}(G,\mathcal{K};B)$
• Figure 5: Configuration as in the proof of Proposition \ref{['prop:CE=2acyclic']} (general case)

Theorems & Definitions (62)

• Lemma 2.1
• proof
• Definition 2.2: $X$-generated group with content function
• Definition 2.3: group reflecting the structure of a Cayley graph
• Theorem 2.4
• proof
• Lemma 2.5: main lemma
• Remark 2.6
• Definition 3.1
• Definition 3.2