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 $\omega$, and an isometric embedding $\iota$ of $M$ to the asymptotic cone ${\mathrm{Cone}}_{\omega }(G)$ such that the induced homomorphism ${\iota }^{\ast }:{\pi }_1(M)\to {\pi }_1({\mathrm{Cone}}_{\omega }(G))$ 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.

This is a corrected version of the paper. Some proofs are improved and several typos are corrected. The main result remains unchanged