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

TL;DR

This paper constructs finite groups that simulate key properties of free groups related to connectivity and relations, and applies this to prove every finite inverse monoid has a finite $F$-inverse cover, solving a longstanding problem.

Contribution

It introduces a method to approximate free group features in finite groups and proves the existence of finite $F$-inverse covers for all finite inverse monoids.

Findings

Finite groups can encode connectivity properties of graphs.

Every finite inverse monoid admits a finite $F$-inverse cover.

The construction provides new insights into free group approximations.

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.