Asymptotic cones of finitely presented groups

Abstract

Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that R-rank(S) is at least 2, and let $Γ$ be a uniform lattice in G. (a) If $CH$ holds, then $Γ$ has a unique asymptotic cone up to homeomorphism. (b) If $CH$ fails, then $Γ$ has $2^{2^ω}$ asymptotic cones up to homeomorphism.