As a part of Gromov’s grand program to study infinite groups as geometric objects, a fundamental, open question is to determine algebraic properties of groups that are quasi-isometry invariants. The seminal work of Gromov on groups of polynomial growth implies that virtual nilpotence is a quasi-isometry invariant property for finitely generated groups. This does not carry over to solvable groups. Nevertheless, the question remains open for polycyclic groups. This is a conjecture which can be rephrased in terms of Lie groups as follows:
Conjecture. Any group quasi-isometric to a lattice in a simply connected solvable Lie group is also virtually a lattice in such a group.
In this survey article, the authors discuss the methods used and results proved towards this conjecture. A spectacular result in this direction was obtained by the authors together with Kevin Whyte when they proved the conjecture for the group Sol, for which the conjecture was made by Farb & Mosher. The group Sol is the solvable Lie group which is a semidirect product of \(\mathbb R\) and \(\mathbb R^2\) where \(z \in\mathbb R\) acts via a diagonal matrix with entries \(e^{z/2}\) and \(e^{-z/2}\).
Irine Peng proved a far-reaching generalization of this theorem in her Ph.D. thesis written under the supervision of the first author. The conjecture is proved for a unimodular, abelian-by-abelian group \(\mathbb R^k \propto\mathbb R^n\) where the action of \(\mathbb R^k\) is by a representation \(\rho : \mathbb R^k \rightarrow \text{SL}(\mathbb R^n)\). The authors believe that the techniques developed to prove the above theorem will provide a way to attack the conjecture in general.
The new technique developed is called ‘coarse differentiation’. As quasi-isometries have no local structure, the conventional derivatives cannot be defined and the definition of coarse derivative models the large scale behaviour of the quasi-isometry. In particular, the authors obtain a coarse analogue of Rademacher’s theorem which shows that a bi-Lipschitz map of \(\mathbb R^n\) is differentiable almost everywhere.
Another interesting question is to determine, given a Lie group, whether there is a finitely generated group quasi-isometric to it. The authors show that the solvable Lie group \(\mathbb R\propto\mathbb R^2\) where \(\mathbb R\) acts by \(z.(x,y) := (e^{az}x,e^{-bz}y)\) for \(a,b>0\), \(a \neq b\), admits no finitely generated group quasi-isometric to it.
The paper outlines an important technique which should prove widely influential. In the statement of Theorem 1.4 on the conjecture for the group \(\mathbb R^k \propto\mathbb R^n\), it seems that the hypothesis that \(\Gamma\) is finitely generated is inadvertently missing.
For the entire collection see [Zbl 1220.00033].