  It is an open question (Pawlikowski) whether every finitely generated group
can be realized as a fundamental group of a compact metric space. In this paper
we prove that any countable group can be realized as the fundamental group of a
compact subspace of four dimensional Euclidean space. According to theorems of
Shelah (see also Pawlikowski) such space can not be locally path connected if
the group is not finitely generated. This constructions complements realization
of groups in the context of compact Hausdorff spaces, that was studied by
Keesling and Rudyak, and Przezdziecki .
