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](http://www.mathjournals.org/mrl/1997-004-005/1997-004-005-008.pdf) 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".



Groups quasi-isometric to reducible nonuniform lattices
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. 


