2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70756We 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.This is a corrected version of the paper. Some proofs are improved and several typos are corrected. The main result remains unchangedGroup TheoryPrimary: 20F69; Secondary: 14F35, 20F06text