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

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 $u⟶v$ in $\mathcal{E}$ then there exists a $G$-equivalent word $q$ which also forms a path $u⟶v$ 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.