Cornulier’s conjecture on quasiisometric classification of completely solvable Lie groups

Prove that any two quasiisometric completely solvable Lie groups are isomorphic. In particular, determine whether this holds even within the subclass of simply connected nilpotent Lie groups, thereby settling the quasiisometric classification in those settings.

Background

The paper studies quasiisometric classifications of Lie groups and emphasizes reduction procedures that compare more tractable models via sublinear bilipschitz equivalence. In this context, Cornulier formulated a conjecture asserting that quasiisometry between completely solvable Lie groups should imply isomorphism, which would provide a definitive classification within this broad class.

Despite significant progress on invariants (e.g., cone-dimensions, Dehn functions, reductions to classes (C0), (C1), (C∞)), the conjecture remains unresolved and, as noted by the authors, is open even when restricted to simply connected nilpotent Lie groups, where Pansu’s results provide partial structure but do not yield a full classification.