arXiv math/05063612005-06math.GRmath.FA

Property (T) and rigidity for actions on Banach spaces

U. Bader, A. Furman, T. Gelander, N. Monod

备注：: Many minor improvements
发表信息：: See final version in: Acta Math 198 No. 1 (2007) 57--105
DOI：: 10.1007/s11511-007-0013-0

英文摘要：

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 $1\leq p<\infty$. We show that the fixed point property for $L^p$ follows from property (T) when $1<p< 2+\e$. For simple Lie groups and their lattices, we prove that the fixed point property for $L^p$ holds for any $1< p<\infty$ 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 Banach spaces.

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