Fundamental groups of asymptotic cones

Details Fundamental groups of asymptotic cones Fundamental groups of asymptotic cones

2004-04-05

arxiv.org/abs/math/0404111v2

Mathematics

Group Theory

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

Scientific paper

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 ${\rm Cone}_\omega (G)$ such that the induced homomorphism $\iota ^ \ast :\pi_1(M)\to \pi_1({\rm Cone}_\omega (G))$ 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.

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