  Let $\Gamma_g$ be a surface group of genus $g\geq 2$. It is known that the
canonical central extension $\tilde{\Gamma}_g$ and the direct product
$\Gamma_g\times \mathbb{Z}$ are quasi-isometric. It is also easy to see that
they are measure equivalent. By contrast, in this paper, we prove that
quasi-isometry and measure equivalence cannot be achieved "in a compatible
way". More precisely, these two groups are not uniform (nor even integrable)
measure equivalent. In particular, they cannot act continuously, properly and
cocompactly by isometries on the same proper metric space, or equivalently they
are not uniform lattices in a same locally compact group.
