Integrable measure equivalence and the central extension of surface groups

TL;DR

We address the compatibility between quasi-isometry and measure equivalence for the central extension of a genus g surface group and its direct product with Z. We prove that tildeΓ_g and Γ_g×Z are not IME nor uniform in a common locally compact group, despite being ME and QI, though an $L^p$-ME coupling exists for every $p<1$; this reveals a sharp obstruction to aligning geometric and measure-theoretic notions of equivalence. The argument combines bounded cohomology and Monod-Shalom rigidity with an ergodic theorem for integrable cocycles to detect nontrivial center-extensions via sublinear growth. These results illuminate the limits of rigidity phenomena for central extensions and provide a framework for understanding how ergodic and geometric properties interact in measure equivalence.

Abstract

Let $Γ_g$ be a surface group of genus $g\geq 2$. It is known that the canonical central extension $\tildeΓ_g$ and the direct product $Γ_g\times \mathbb{Z}$ 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.

Theorems & Definitions (28)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Lemma 3.1
• Theorem 3.2
• Proposition 3.3
• proof
• Proposition 3.4