  The first author has associated in a natural way a profinite group to each
irreducible subshift. The group in question was initially obtained as a maximal
subgroup of a free profinite semigroup. In the case of minimal subshifts, the
same group is shown in the present paper to also arise from geometric
considerations involving the Rauzy graphs of the subshift. Indeed, the group is
shown to be isomorphic to the inverse limit of the profinite completions of the
fundamental groups of the Rauzy graphs of the subshift. A further result
involving geometric arguments on Rauzy graphs is a criterion for freeness of
the profinite group of a minimal subshift based on the Return Theorem of
Berth\'e et. al.
