Property ($T$) and rigidity for actions on Banach spaces

Uri Bader Technion–Israel Institute of Technology, Technion City, Haifa, Israel Alex Furman University of Illinois at Chicago, Department of Mathematics, Statistics and Computer Science, 851 South Morgan Street, Chicago, IL, USA Tsachik Gelander The Hebrew University, Institute of Mathematics, Givat Ram, Jerusalem, Israel Nicolas Monod Université de Genève, Section de Mathématiques, Case postale 64, 2-4, rue du Livre, Genève 4, Switzerland

TBD mathscidoc:1701.331980

Acta Mathematica, 198, (1), 57-105, 2005.8
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.
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  title={Property ($T$) and rigidity for actions on Banach spaces},
  author={Uri Bader, Alex Furman, Tsachik Gelander, and Nicolas Monod},
  booktitle={Acta Mathematica},
Uri Bader, Alex Furman, Tsachik Gelander, and Nicolas Monod. Property ($T$) and rigidity for actions on Banach spaces. 2005. Vol. 198. In Acta Mathematica. pp.57-105.
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