On irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups

TL;DR

This paper investigates the irreducibility and disjointness of Koopman and quasi-regular representations of weakly branch groups acting on the boundary of a rooted tree, revealing conditions for irreducibility and disjointness.

Contribution

It establishes that Koopman representations are irreducible under certain measures and that different quasi-regular and Koopman representations are pairwise disjoint, expanding understanding of these representations.

Findings

Koopman representations are irreducible with quasi-invariant Bernoulli measures.

Quasi-regular and Koopman representations corresponding to different orbits or measures are disjoint.

The centralizer of G in transformation groups on the boundary is trivial.

Abstract

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.