- journal article metadata
European Mathematical Society Publishing House
2016-09-19 17:05:02
Groups, Geometry, and Dynamics
Groups Geom. Dyn.
GGD
1661-7207
1661-7215
Group theory and generalizations
10.4171/GGD
http://www.ems-ph.org/doi/10.4171/GGD
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
10
2016
3
Integrable measure equivalence and the central extension of surface groups
Kajal
Das
École Normale Supérieure de Lyon, LYON CEDEX 07, FRANCE
Romain
Tessera
Université Paris-Sud, CNRS, ORSAY CEDEX, FRANCE
Integrable measure equivalence, quasi-isometry, central extension, surface groups
Let $\Gamma_g$ be a surface group of genus $g \geq 2$. It is known that the canonical central extension $\wtilde{\Gamma}_g$ 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.
Group theory and generalizations
Dynamical systems and ergodic theory
Geometry
965
983
10.4171/GGD/373
http://www.ems-ph.org/doi/10.4171/GGD/373