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Property \((T)\) and rigidity for actions on Banach spaces. (English) Zbl 1162.22005
Classically, a locally compact second countable group \(G\) has Kazhdan’s property (T) when any unitary representation of \(G\), with almost invariant vectors, has automatically nontrivial invariant vectors. This is known to be equivalent to Serre’s fixed point property (FH) for affine isometric actions on Hilbert spaces. The work under review undertakes a systematic study of analogue properties (T\(_B\)) and (F\(_B\)) where Hilbert spaces are being replaced by Banach spaces \(B\).
The first part of the paper investigates relations between properties (T), (T\(_B\)) and (F\(_B\)) for general locally compact second countable groups when \(B=L^p (\mu)\), with \(\sigma\)-finite measure \(\mu\) on a standard Borel measure space \((X,{\mathcal B},\mu)\). They are summarized in the following two results: 6mm
I.
(T) implies (T\(_B\)) when \(B=L^p(\mu)\), \(1\leqslant p<\infty\), or \(B\) is a closed subspace of \(L^p(\mu)\), \(1<p<\infty\), \(p\neq 4,6,8,\ldots\), or \(B\) is a quotient space of \(L^p(\mu)\), \(1<p<\infty\), \(p\neq \frac{4}{3},\frac{6}{5},\frac{8}{7},\ldots\) Conversely, if (T\(_{L^p[0,1]}\)) holds for some \(1<p<\infty\), then (T) holds.
II.
(F\(_B\)) implies (T\(_B\)) for any Banach space \(B\). In the opposite direction, (T) implies (F\(_B\)) whenever \(B\) is a closed subspace of \(L^p(\mu)\), \(1<p< 2+\varepsilon\), where the constant \(\varepsilon=\varepsilon (G)>0\) might depend on the group \(G\).
The second part is concerned with examples of groups that satisfy (F\(_{L^p}\)). It is proved that a large class of higher-rank groups \(G={\mathbf G}_1(k_1)\times \cdots \times {\mathbf G}_m(k_m)\) and their lattices have property (F\(_B\)) for all Banach spaces \(B\) mentioned in I. It is also conjectured that these groups should satisfy some analogous properties (\(\overline{\text{T}}_B\)) and (\(\overline{\text{F}}_B\)) for all superreflexive topological vector spaces \(B\). In this direction, two splitting results about uniformly equicontinuous affine actions on superreflexive spaces of topological group products \(G=G_1\times \cdots \times G_n\), and respectively of their lattices, are proved. The Appendix contains a generalization, proved by Y. Shalom, of the Howe-Moore theorem on the vanishing of matrix coefficients for unitary representations, to the situation of uniformly convex and uniformly smooth Banach spaces.

MSC:
22D40 Ergodic theory on groups
20G15 Linear algebraic groups over arbitrary fields
22D10 Unitary representations of locally compact groups
22D12 Other representations of locally compact groups
22E40 Discrete subgroups of Lie groups
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
53C24 Rigidity results
58D19 Group actions and symmetry properties
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