Integrable measure equivalence and the central extension of surface groups

Let ${\mathrm{\Gamma }}_g$ be a surface group of genus $g\ge 2$. It is known that the canonical central extension ${\tilde{\mathrm{\Gamma }}}_g$ and the direct product ${\mathrm{\Gamma }}_g\times \mathbb{Z}$ 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.