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 w, and an isometric embedding i of M to the asymptotic cone Cone(w)(G) such that the induced homomorphism i* : pi(1) (M) -> pi(1) (Cone, (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. (c) 2005 Elsevier Ltd. All rights reserved.