Asymptotic cones of Lie groups and cone equivalences

TL;DR

This paper introduces cone bilipschitz equivalences, a new concept extending quasi-isometries, to analyze asymptotic cones of Lie groups, revealing reductions to solvable and nilpotent cases for better understanding.

Contribution

It defines cone bilipschitz equivalences and applies them to simplify the study of asymptotic cones of Lie groups, especially solvable and nilpotent groups.

Findings

Cone bilipschitz equivalences induce bilipschitz homeomorphisms between asymptotic cones.

Asymptotic cones of connected Lie groups can be studied via those of solvable Lie groups.

The approach provides new insights into the structure of nilpotent groups' asymptotic cones.

Abstract

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.