Acta Mathematica

Property (T) and rigidity for actions on Banach spaces

Uri Bader, Alex Furman, Tsachik Gelander, and Nicolas Monod

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We study property (T) and the fixed-point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for Lp follows from property (T) when 1< p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for Lp holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces.


Bader partially supported by ISF grant 100146; Furman partially supported by NSF grants DMS-0094245 and DMS-0604611; Gelander partially supported by NSF grant DMS-0404557 and BSF grant 2004010; Monod partially supported by FNS (CH) and NSF (US).

Article information

Acta Math. Volume 198, Number 1 (2007), 57-105.

Received: 2 August 2005
Accepted: 5 February 2007
First available in Project Euclid: 31 January 2017

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2007 © Institut Mittag-Leffler


Bader, Uri; Furman, Alex; Gelander, Tsachik; Monod, Nicolas. Property ( T ) and rigidity for actions on Banach spaces. Acta Math. 198 (2007), no. 1, 57--105. doi:10.1007/s11511-007-0013-0.

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