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TBDmathscidoc: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.
```@inproceedings{uri2005property,
title={Property (\$T\$) and rigidity for actions on Banach spaces},
author={Uri Bader, Alex Furman, Tsachik Gelander, and Nicolas Monod},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203350215148689},
booktitle={Acta Mathematica},
volume={198},
number={1},
pages={57-105},
year={2005},
}
```
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. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203350215148689.