Geodesic
Integrable measure equivalence and the central extension of surface groups
Kajal Das1 ; Romain Tessera2
1École Normale Supérieure de Lyon, France
2Université Paris-Sud, Orsay, France
Groups, geometry, and dynamics, Tome 10 (2016) no. 3, pp. 965-983

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DOI

Let Γg​ be a surface group of genus g≥2. It is known that the canonical central extension Γg​ and the direct product Γg​×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.
DOI : 10.4171/ggd/373
Classification : 20-XX, 37-XX, 51-XX
Mots-clés : Integrable measure equivalence, quasi-isometry, central extension, surface groups

Kajal Das  1   ; Romain Tessera  2

1 École Normale Supérieure de Lyon, France
2 Université Paris-Sud, Orsay, France 
Kajal Das; Romain Tessera. Integrable measure equivalence and the central extension of surface groups. Groups, geometry, and dynamics, Tome 10 (2016) no. 3, pp. 965-983. doi: 10.4171/ggd/373
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