Abstract. The Haagerup property for locally compact groups is generalised to the context of locally compact quantum groups, with several equivalent characterisations in terms of the unitary representations and positive-definite functions established. In particular it is shown that a locally compact quantum group G has the Haagerup property if and only if its mixing representations are dense in the space of all unitary representations. For discrete G we characterise the Haagerup property by the existence of a symmetric proper conditionally negative functional on the dual quantum group O G; by the existence of a real proper cocycle on G, and further, if G is also unimodular we show that the Haagerup property is a von Neumann property of G. This extends results of Akemann, Walter, Bekka, Cherix, Valette, and Jolissaint to the quantum setting and provides a connection to the recent work of Brannan. We use these characterisations to show that the Haagerup property is preserved under free products of discrete quantum groups.
We give a simple definition of property T for discrete quantum groups, and prove the basic expected properties: discrete quantum groups with property T are finitely generated and unimodular. Moreover we show that, for "I.C.C." discrete quantum groups, property T is equivalent to Connes' property T for the dual von Neumann algebra. This allows us to give the first example of a property T discrete quantum group which is not a group using the twisting construction.
We show that a group acting on a non-trivial tree with finite edge stabilizers and icc vertex stabilizers admits a faithful and highly transitive action on an infinite countable set. This result is actually true for infinite vertex stabilizers and some more general, finite of infinite, edge stabilizers that we call highly core-free. We study the notion of highly core-free subgroups and give some examples. In the case of amalgamated free products over highly core-free subgroups and HNN extensions with highly core-free base groups we obtain a genericity result for faithful and highly transitive actions. In particular, we recover the result of D. Kitroser stating that the fundamental group of a closed, orientable surface of genus g > 1 admits a faithful and highly transitive action.An action of a countable group Γ on an infinite countable set X is called highly transitive if it is n-transitive for all n ≥ 1. It is easy to see that an action Γ X is highly transitive if and only if the image of Γ in the Polish group S(X) of bijections of X is dense.An obvious example of a highly transitive and faithful action is given by the action of the countable group of finitely supported permutations of X. This group is not finitely generated and far from being free (it is amenable).The first explicit construction of a highly transitive and faithful action of the free group F n , for 2 ≤ n ≤ ∞, was published in [McD76] by T.P. McDonough. Then, J. D. Dixon [Di89] showed that most (in a topological sense) finitely generated highly transitive subgroups of S(X) are free.A.M.W. Glass and S.H. McCleary [GM90] have constructed faithful and highly transitive action of a free product Γ 1 * Γ 2 of non-trivial countable (or finite) groups with Γ 2 having an element of infinite order. They also observed that Z 2 * Z 2 does not have a faithful and 2-transitive action and they asked for which non-trivial groups Γ i , i = 1, 2, does Γ 1 * Γ 2 have a faithful and highly transitive action.S.V. Gunhouse [Gu91] completely answered the question by constructing, for any non-trivial countable (or finite) groups Γ i , i = 1, 2, with one of the Γ i of size at least 3, a faithful and highly transitive action of Γ 1 * Γ 2 . A similar result was obtained independently by K.K. Hickin [Hi90]. Recently, the last two authors [MS12] proved a genericity result for faithful and highly transitive actions of free products.Using the techniques of Hickin, Gunhouse also characterized in his PhD (unpublished result) the non-trivial amalgamated free products with amalgamation over an Artinian group 1 , that admit a faithful and highly transitive action. He proved that if Γ = Γ 1 * Σ Γ 2 and Σ is an Artinian group, properly included in Γ 1 and Γ 2 , then Γ admits a faithful and highly transitive action The first author is partially supported by ANR Grants OSQPI and NEUMANN. The second author is partially supported by the Conseil Regional de Bourgogne (Faber 2012-1-9201-247). 1 that is, any decreasing chain of its distinct subgroups terminates after a finite number; e.g. ...
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