Property (<Emphasis Type="Italic">T</Emphasis>) and rigidity for actions on Banach spaces |
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Authors: | Uri Bader Alex Furman Tsachik Gelander Nicolas Monod |
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Institution: | (1) Technion–Israel Institute of Technology, Technion City, Haifa, 32000, Israel;(2) University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science, 851 South Morgan Street, Chicago, IL 60607-7045, USA;(3) The Hebrew University, Institute of Mathematics, Givat Ram, Jerusalem, 91904, Israel;(4) Université de Genève, Section de Mathématiques, 2-4, rue du Livre, Case postale 64, CH-1211 Genève 4, Switzerland |
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Abstract: | We study property (T) and the fixed-point property for actions on L
p
and other Banach spaces. We show that property (T) holds when L
2 is replaced by L
p
(and even a subspace/quotient of L
p
), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for L
p
follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for L
p
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). |
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Keywords: | |
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