Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees

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 ...

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...

In this paper we investigate relations between Koopman, groupoid and quasi-regular representations of countable groups. We show that for an ergodic measure class preserving action of a countable group G on a standard Borel space the associated groupoid and quasi-regular representations are weakly equivalent and weakly contained in the Koopman representation. Moreover, if the action is hyperfinite then the Koopman representation is weakly equivalent to the groupoid. As a corol...

We consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree. We introduce and discuss the new notion of locally 2-transitive actions. Assuming local 2-transitivity our main theorem yields a precise decomposition of the boundary representation into irreducible constituents. The method can be used to study Gelfand pairs and enables us to answer a q...

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that non-amenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up...

We prove a general result about the decomposition on ergodic components of group actions on boundaries of spherically homogeneous rooted trees. Namely, we identify the space of ergodic components with the boundary of the orbit tree associated with the action, and show that the canonical system of ergodic invariant probability measures coincides with the system of uniform measures on the boundaries of minimal invariant subtrees of the tree. A special attention is given to the ...

We introduce notions of absolutely non-free and perfectly non-free group actions and use them to study the associated unitary representations. We show that every weakly branch group acts absolutely non-freely on the boundary of the associated rooted tree. Using this result and the symmetrized diagonal actions we construct for every countable branch group infinitely many different ergodic perfectly non-free actions, infinitely many II$_1$-factor representations, and infinitely...

Consider an infinite homogeneous tree $T_n$ of valence $n+1$, its group $Aut(T_n)$ of automorphisms, and the group $Hie(T_n)$ of its spheromorphisms (hierarchomorphisms), i.~e., the group of homeomorphisms of the boundary of $T_n$ that locally coincide with transformations defined by automorphisms. We show that the subgroup $Aut(T_n)$ is spherical in $Hie(T_n)$, i.~e., any irreducible unitary representation of $Hie(T_n)$ contains at most one $Aut(T_n)$-fixed vector. We presen...

We show that the ordinary irreducible representations of the group of automorphisms of a finite, rooted tree may be obtained following one of the classical method used for the irreducible representations of the symmetric group, namely the approach in Chapter 2 of the classical book by James and Kerber. To reach this goal, we introduce and study the combinatorics of tree compositions, a natural generalization of set compositions but with new features and more complexity due to...

Let $G$ be a discrete group with property (T). It is a standard fact that, in a unitary representation of $G$ on a Hilbert space $\mathcal{H}$, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation $\sigma$, then the vector is close to a sub-representation isomorphic to $...