Integrable measure equivalence and the central extension of surface groups

TL;DR

This paper investigates the relationship between quasi-isometry and measure equivalence for surface groups and their central extensions, showing they cannot be compatible in a strong sense, revealing new distinctions in geometric group theory.

Contribution

It proves that the canonical central extension of surface groups and their direct product are not uniformly measure equivalent, despite being quasi-isometric and measure equivalent in a weaker sense.

Findings

The groups are not uniform measure equivalent.

They cannot act properly and cocompactly on the same metric space.

They are not uniform lattices in the same locally compact group.

Abstract

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.