Asymptotic cones of Lie groups and cone equivalences

We introduce cone bilipschitz equivalences between metric spaces. These are maps, more general than quasi-isometries, that induce a bilipschitz homeomorphism between asymptotic cones. Non-trivial examples appear in the context of Lie groups, and we thus prove that the study of asymptotic cones of connected Lie groups can be reduced to that of solvable Lie groups of a special form. We also focus on asymptotic cones of nilpotent groups.