Fundamental groups of asymptotic cones

Abstract

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 ${\rm Cone}_ω(G)$ such that the induced homomorphism $ι^ \ast :π_1(M)\to π_1({\rm Cone}_ω(G))$ 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.