Open Access
2007 Property (T) and rigidity for actions on Banach spaces
Uri Bader, Alex Furman, Tsachik Gelander, Nicolas Monod
Author Affiliations +
Acta Math. 198(1): 57-105 (2007). DOI: 10.1007/s11511-007-0013-0


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.

Funding Statement

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).


Download Citation

Uri Bader. Alex Furman. Tsachik Gelander. Nicolas Monod. "Property (T) and rigidity for actions on Banach spaces." Acta Math. 198 (1) 57 - 105, 2007.


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

zbMATH: 1162.22005
MathSciNet: MR2316269
Digital Object Identifier: 10.1007/s11511-007-0013-0

Rights: 2007 © Institut Mittag-Leffler

Vol.198 • No. 1 • 2007
Back to Top