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Register a new userAutomorphism groups of Coxeter groups do not have Kazhdans property (T)
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Authors
Olga Varghese
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We show that for a large class $mathcal{W}$ of Coxeter groups the following holds: Given a group $W_Gamma$ in $mathcal{W}$, the automorphism group ${rm Aut}(W_Gamma)$ virtually surjects onto $W_Gamma$. In particular, the group ${rm Aut}(G_Gamma)$ is virtually indicable and therefore does not satisfy Kazhdans property (T). Moreover, if $W_Gamma$ is not virtually abelian, then the group ${rm Aut}(W_Gamma)$ is large.
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We show that the automorphism group of a graph product of finite groups $Aut(G_Gamma)$ has Kazhdans property (T) if and only if $Gamma$ is a complete graph.
We give a partial solution to a long-standing open problem in the theory of quantum groups, namely we prove that all finite-dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups (Admissibility Conjecture for quantum group representations). We use this to study Kazhdans Property (T) for quantum groups with non-trivial scaling group, strengthening and generalising some of the earlier results obtained by Fima, Kyed and So{l}tan, Chen and Ng, Daws, Skalski and Viselter, and Brannan and Kerr. Our main results are: (i) All finite-dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible. (ii) A generalisation of a theorem of Wang which characterises Property (T) in terms of isolation of finite-dimensional irreducible representations in the spectrum. (iii) A very short proof of the fact that quantum groups with Property (T) are unimodular. (iv) A generalisation of a quantum version of a theorem of Bekka--Valette proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of non-existence of almost invariant vectors for weakly mixing representations. (v) A generalisation of a quantum version of Kerr-Pichot theorem, proven earlier for quantum groups with trivial scaling group, which characterises Property (T) in terms of denseness properties of weakly mixing representations.
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