Property (T) and rigidity for actions on Banach spaces

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 $1\le p<\infty$. 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.

Many minor improvements

References in corpus (1)

• Superrigidity for irreducible lattices and geometric splitting

Cited by in corpus (30)

• Nonlinear spectral calculus and super-expanders
• Poincaré inequalities, embeddings, and wild groups
• A fixed point theorem for L^1 spaces
• Isometric group actions on Banach spaces and representations vanishing at infinity
• Strong property (T) for higher rank simple Lie groups
• Approximation properties for noncommutative $L^p$-spaces of high rank lattices and nonembeddability of expanders
• Fixed point properties and second bounded cohomology of universal lattices on Banach space
• Proper isometric actions of hyperbolic groups on $L^p$-spaces
• Strong Banach Property (T) for Simple Algebraic Groups of Higher Rank
• A finitely generated, locally indicable group with no faithful action by C^1 diffeomorphisms of the interval
• Random groups, random graphs and eigenvalues of p-Laplacians
• Towards Strong Banach property (T) for SL(3,R)
• On isometry groups and maximal symmetry
• Acylindrically hyperbolic groups with exotic properties
• Banach space actions and $L^2$-spectral gap
• From the geometry of box spaces to the geometry and measured couplings of groups
• Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings
• Stable actions of central extensions and relative property (T)
• Product groups acting on manifolds
• On the structure and arithmeticity of lattice envelopes
• Spectral gap in the group of affine transformations over prime fields
• Automorphism groups of simplicial complexes and rigidity for uniformly bounded representations
• An average John theorem
• Sphere equivalence, Banach expanders, and extrapolation
• Amenability versus non-exactness of dense subgroups of a compact group
• Proper actions on ${\ell }^p$ spaces for relatively hyperbolic groups
• Property FW, differentiable structures, and smoothability of singular actions
• Banach property (T) for $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{Z})$ and its applications
• Euclidean-valued group cohomology is always reduced
• Group 1-cohomology is complemented