  We exhibit a variety of groups that act properly and even cocompactly on
median graphs (a.k.a. one-skeletons of CAT(0) cube complexes), with
quasi-isometric groups that do not admit any proper action on a median graph.
This answes a question of Niblo, Sageev and Wise. Our examples are all
quasi-isometrically trivial central extensions of certain cubulated groups.
