Quasi-isometric nilpotent groups isomorphism problem

Determine whether any two quasi-isometric nilpotent Lie groups are isomorphic. Specifically, establish whether quasi-isometry implies isomorphism for simply connected nilpotent Lie groups, thereby resolving the quasi-isometric classification problem in this setting.

Background

The paper studies invariants of nilpotent Lie groups under Lipschitz and quasi-isometric maps and extends cohomological invariance results. A central classification question in geometric group theory asks whether quasi-isometry determines the group up to isomorphism. For nilpotent groups, Pansu’s theorem ties large-scale geometry to Lie algebraic data via asymptotic cones, and Shalom–Sauer–Gotfredsen–Kyed showed cohomology algebra invariance under quasi-isometry. Despite these advances, the core question of whether quasi-isometric nilpotent groups must be isomorphic is not settled.