Isometric Group Actions on Hilbert Spaces: Growth of Cocycles

By Yves Cornulier, Romain Tessera, Alain Valette. 2005

106 citations recorded by the source corpus.

Abstract.We study growth of 1-cocycles of locally compact groups, with values in unitary representations. Discussing the existence of 1-cocycles with linear growth, we obtain the following alternative for a class of amenable groups G containing polycyclic groups and connected amenable Lie groups: either G has no quasi-isometric embedding into a Hilbert space, or G admits a proper cocompact action on some Euclidean space. On the other hand, noting that almost coboundaries (i.e. 1-cocycles approximable by bounded 1-cocycles) have sublinear growth, we discuss the converse, which turns out to hold for amenable groups with “controlled” Følner sequences; for general amenable groups we prove the weaker result that 1-cocycles with sufficiently small growth are almost coboundaries. Besides, we show that there exist, on a-T-menable groups, proper cocycles with arbitrary small growth.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8c64697f0ca6f9d81e91f0e2667f44782f5ae7ab

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Citations in the corpus

Speed of random walks, isoperimetry and compression of finitely generated groups (2015, unknown): “1112/jtopol/jtt019 [26] de Cornulier, Yves; Tessera, Romain; Valette, Alain, Isometric group actions on {H}ilbert spaces: growth of cocycles, Geom.”
Lp compression, traveling salesmen, and stable walks (2009, unknown): “⊳”
RELATIVE KAZHDAN PROPERTY (2005, background): “for a polycyclic group Γ, the H-metric is equivalent to the word metric, although it is known that Γ has no conditionally negative deﬁnite function with quadratic growth unless Γ is virtually abelian [CTV]. 3. Relative Property (T) in connected Lie groups and p-adic algebraic groups 3.1. Preliminari”
Embeddings of Discrete Groups and the Speed of Random Walks (2007, methodology): “Hereβ∗(G) is the largestβ ≥ 0 for which there exists a set of generatorsS of G andc > 0 such that for all t ∈ N we haveE[dG(Wt, e) ] ≥ ctβ, where{Wt}∞t=0 is the canonical simple random walk on the Cayley graph ofG determined byS, starting at the identity element.”
On the joint spectral radius for isometries of non-positively curved spaces and uniform growth (2018, unknown): “Hilbert compression as in [34] ; it is ubiquitous in the study of free group automorphisms and it is key to constructing limits of representations, e.”
Smooth Fourier multipliers on group von Neumann algebras (2010, result): “lead us to ﬁnd a natural framework for Caldero´n-Zygmund theory in the free group algebra. Regarding other discrete groups, we refer to [68] for recent results on group cocycles when Hψ= ℓ2(G) and to [17, 18, 92] for other related results. 5.2. Hormander-Mihlin dimension. According to our geometrica”