Fundamental groups of asymptotic cones

We show that for any metric space M satisfying certain natural conditions, there is a finitely generated group G, an ultrafilter ω , and an isometric embedding ι of M to the asymptotic cone Cone ω ( G ) such that the induced homomorphism ι * : π 1 ( M ) → π 1 ( Cone ω ( 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.