For a finite connected graph 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 EβͺEβ1) is closed under deletion
of generators (i.e.~edges). As a consequence, every element gβG admits a
unique minimal set C(g) of edges (the \emph{content} of g) needed
to represent g as a word over C(g)βͺC(g)β1. The
crucial property of the group G is that connectivity in the graph
E is encoded in G in the following sense: if a word p forms a
path uβΆv in E then there exists a G-equivalent
word q which also forms a path uβΆv and uses only edges from
their content; in particular, the content of the corresponding group element
[p]Gβ=[q]Gβ spans a connected subgraph of E containing the
vertices u and v. 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.Comment: 46 pages, 14 figures, new result Cor. 2.7 included, several
inaccuracies removed, more details include