TITLE: Groups quasi-isometric to reducible nonuniform lattices
QUESTION [7 upvotes]: It is known that a finitely group $G$ is quasi-isometric to a nonuniform  irreducible lattice $\Lambda$ in a semisimple Lie group if and only if $G$ and
$\Lambda$ are commensurable (see references in   this survey of Farb). 
 Question. What is known about groups quasi-isometric to reducible nonuniform lattices in semisimple Lie groups? 
As usual in this business "semisimple" means "noncompact, connected, semisimple, with finite center".

REPLY [6 votes]: Here is a partial answer: Suppose $\Gamma = \Gamma_1 \times \dots \times \Gamma_n$ and all the $\Gamma_i$ are irreducible lattices in $G_i$, where each $G_i$ has real rank at least two.  
It has been a long time, and I do not remember all the details, but I think it may be true that any quasi-isometry from a product of such lattices $\Gamma_1 \times \dots \times \Gamma_n$ to itself preserves the factors (up to permutation). 
I am looking at Lemma 10.3 of my paper in JAMS from 1998 http://www.math.uchicago.edu/~eskin/sl3z.ps.  It is stated for irreducible lattices, but that does seem to be used in the proof. Of course I could be missing something.
If self quasi-isometries are indeed factor preserving, then one has the same classification as for irreducible lattices.
One more comment: the reason my proof fails when you have a factor $\Gamma_i$ in a real rank one group $G_i$ is that I quote Lubotsky-Mozes-Raghunathan which does not work in that case.