Property (T) and rigidity for actions on Banach spaces
By U. Bader, A. Furman, Tsachik Gelander, Nicolas Monod. 2005
220 citations recorded by the source corpus.
We study property (T) and the fixed-point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for Lp follows from property (T) when 1<p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for Lp 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.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/965b294bb6946a839e634fdffba3720333fd37da
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Citations in the corpus
- GROUP ACTIONS ON 1-MANIFOLDS: A LIST OF VERY CONCRETE OPEN QUESTIONS (2017, background): “ilbert space has an invariant vector. (See [7, 92] for a full discussion on this.) 3 Property (T) has very strong consequences for the dynamics of group actions in dierent settings; see for instance [6, 48, 119, 125, 138]. In what concerns actions on 1-dimensional spaces, a classical result pointing”
- A fixed point theorem for L1 spaces (2010, methodology): “Theorem A in [1] shows that (TL1) implies (T ); although it is claimed there for L with 1 < p < ∞, the proof applies unchanged to L1, using the Connes–Weiss construction [4] (exposed also in Theorem 6.”
- Nonlinear spectral calculus and super-expanders (2012, background): “Lafforgue [29] on the strengthened version of property (T) for SL3(F) when F is a non-Archimedian local field (see also [3, 31]).”
- Groups acting on manifolds: around the Zimmer program (2008, result): “A result noticed by the author and Margulis and contained in a paper of Bader, Furman, Gelander, and Monod allows one to adapt this proof for values of k slightly less than 3 2 [5, 81].”
- Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces (2006, background): “ρλ G,p (t) where ρis the compression of b. Remark 3.3. It is not difficult to prove that the asymptotic behavior of Mρλ G,p is invariant under quasi-isometry between finitely generated groups. Following [BFGM], we say that Ghas Property (FLp) if H1(G,π) = 0 for every norm-preserving Lp-representation π”
- Poincaré inequalities, embeddings, and wild groups (2010, background): “See [12, 4] for a discussion of the relation between property (T ) and fixed points of actions on Lp.”
- Strong Banach property (T) for simple algebraic groups of higher rank (2013, methodology): “e [Laf09], [Pis10] and [MN]). We turnto application to fixed-point property. As a consequence of proposition 5.6 in [Laf09], we immediately obtain the following proposition, confirming conjecure 1.6 in [BFGM] for any simple algebraic group of higher rank over a non archimedean local field and its cocom”
- Speed of random walks, isoperimetry and compression of finitely generated groups (2015, methodology): “lemented in X, X= X(Γ) ⊕X′(), and the decomposition is canonical. Definition 6.5. Let Γ be a discrete group equipped with finite generating set S, X be a uniformly convex Banach space. 49 • Following [8], we say that Γ has Property (F X) if any action of Γ on X by affine isometries has a Γ-fixed point. •”