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Register a new userOn irreducibility of Koopman representations of Higman-Thompson groups
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Authors
Artem Dudko
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We introduce a notion of measure contracting actions and show that Koopman representations corresponding to ergodic measure contracting actions are irreducible. As a corollary we obtain that Koopman representations associated to canonical actions of Higman-Thompson groups are irreducible. We also show that the actions of weakly branch groups on the boundaries of rooted trees are measure contracting. This gives a new point of view on irreducibility of the corresponding Koopman representations.
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We study Koopman and quasi-regular representations corresponding to the action of arbitrary weakly branch group G on the boundary of a rooted tree T. One of the main results is that in the case of a quasi-invariant Bernoulli measure on the boundary of T the corresponding Koopman representation of G is irreducible (under some general conditions). We also show that quasi-regular representations of G corresponding to different orbits and Koopman representations corresponding to different Bernoulli measures on the boundary of T are pairwise disjoint. This gives two continual collections of pairwise disjoint irreducible representations of a weakly branch group. Another corollary of our results is triviality of the centralizer of G in various groups of transformations on the boundary of T.
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