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Integrable measure equivalence and the central extension of surface groups. (English) Zbl 1388.22017

Summary: 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.

MSC:

22F10 Measurable group actions
22D12 Other representations of locally compact groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
20F65 Geometric group theory
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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