  We show that for any metric space $M$ satisfying certain natural conditions,
there is a finitely generated group $G$, an ultrafilter $\omega $, and an
isometric embedding $\iota $ of $M$ to the asymptotic cone ${\rm Cone}_\omega
(G)$ such that the induced homomorphism $\iota ^ \ast :\pi_1(M)\to \pi_1({\rm
Cone}_\omega (G))$ is injective. In particular, we prove that any countable
group can be embedded into a fundamental group of an asymptotic cone of a
finitely generated group.
