Weak Amenability of Hyperbolic Groups

By Nathanial P. Brown Narutaka Ozawa. 2007

84 citations recorded by the source corpus.

We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit of Haagerup and prove that for the word length metric d on a hyperbolic group, the Schur multipliers associated with r^d have uniformly bounded norms for 0<r<1. We then combine this with a Bozejko-Picardello type inequality to obtain weak amenability.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ff621a018a969b6ac8e24f8804c532b9ed41e6df

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Citations in the corpus

Classification and rigidity for von Neumann algebras (2012, background): “an subalgebra, up to unitary conjugacy. Without explaining further details, let us mention the three main ingredients of the proof of Theorem 6.3. Firstly, since hyperbolic groups are weakly amenable [Oz07], the conjugation action NM(P) yP is weakly compact, for any diﬀuse von Neumann subalgebra P o”
Examples of groups which are not weakly amenable (2010, methodology): “nability and passes to closed subgroups. It is proved by De Canni`ere–Haagerup, Cowling and Cowling–Haagerup ([dCH, Co, CH]) that real simple Lie groups of real rank one are weakly amenable (see also [Oz]), and by Haagerup ([Ha]) that real simple Lie groups of real rank at least two are not weakly a”
Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups (2014, background): “ on compact operators we get ϕ(uT) = ϕ(Ju∗JT) = lim k ϕ(xkT) = lim k ϕ(Txk) = ϕ(TJu∗J) = ϕ(Tu). Theorem 3.5 ([OP10a, CS13]). If Γ is bi-exact and weakly amenable (this is the case if Γ is hyperbolic, [Oz08]) then LΓ is strongly solid, in the sense that the normalizer of a diﬀuse amenable subalgebra ”
Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups (2012, background): “hyperbolic and therefore belongs to class S, also Γ belongs to class S. From [CH88] we know that lattices in connected noncompact rank one simple Lie groups with ﬁnite center, are weakly amenable. By [Oz07] hyperbolic groups are weakly amenable. The following argument of [Oz12] shows that Sela’s lim”
Type III factors with unique Cartan decomposition (2012, background): “[CS11] showed that L∞(X)⋊Γ has a unique Cartan subalgebra up to unitary conjugacy for all proﬁnite free ergodic pmp actions of all nonamenable, weakly amenable groups Γ in the class QHreg. Because of [Oz07], this covers all nonelementary hyperbolic groups. This result was then extended in [CSU11] to”