Irreducible Koopman representations for nonsingular actions on boundaries of rooted trees

The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand, how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(o...

We show that weak containment of free ergodic measure-preserving actions of $\mathbf{F}_\infty$ is not equivalent to weak containment of the corresponding Koopman representations. This result is based on the construction of an invariant random subgroup of $\mathbf{F}_\infty$ which is supported on the maximal actions.

This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The main theorems relate the zeta function to determinants of operators defined on edges or vertices of the tree. A zeta function associated to a non-uniform tree lattice with appropriate Hilbert representation is defined. Zeta functions are def...

Let G be a branch group (as defined by Grigorchuk) acting on a tree T. A parabolic subgroup P is the stabiliser of an infinite geodesic ray in T. We denote by $\rho_{G/P}$ the associated quasi-regular representation. If G is discrete, these representations are irreducible, but if G is profinite, they split as a direct sum of finite-dimensionalrepresentations $\rho_{G/P_{n+1}}\ominus\rho_{G/P_n}$, where P_n is the stabiliser of a level-n vertex in T. For a few concrete exa...

For any countable group, and also for any locally compact second countable, compactly generated topological group, G, we show the existence of a "universal" hypercyclic (i.e. topologically transitive) representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of G.

The most degenerate unitary principal series representations {\pi}_{i{\lambda},{\delta}} (with {\lambda} \in R, \delta \in Z/2Z) of G = GL(N,R) attain the minimum of the Gelfand-Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction {\pi}_{i{\lambda},{\delta}}|_H (branching law) with respect to all symmetric pairs (G,H). For N=2n with n \geq 2, the restriction {\pi}_{i{\...

We study the construction of quasimorphisms on groups acting on trees introduced by Monod and Shalom, that we call median quasimorphisms, and in particular we fully characterise actions on trees that give rise to non-trivial median quasimorphisms. Roughly speaking, either the action is highly transitive on geodesics, it fixes a point in the boundary, or there exists an infinite family of non-trivial median quasimorphisms. In particular, in the last case the second bounded coh...

We present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group.

We investigate invariant random subgroups in groups acting on rooted trees. Let $\mathrm{Alt}_f(T)$ be the group of finitary even automorphisms of the $d$-ary rooted tree $T$. We prove that a nontrivial ergodic IRS of $\mathrm{Alt}_f(T)$ that acts without fixed points on the boundary of $T$ contains a level stabilizer, in particular it is the random conjugate of a finite index subgroup. Applying the technique to branch groups we prove that an ergodic IRS in a finitary regul...

Consider an infinite tree. A hierarchomorphism (spheromorphism) is a homeomorphism of the absolute which can be extended to the tree except a finite subtree. Examples of groups of hierarchomorphisms: groups of locally analitic diffeomorphisms of $p$-adic line; also Richard Thompson groups. The groups of hierarchomorphisms have some properties similar to the group of diffeomorphisms of the circle. We discuss actions of groups of ierarchomorphisms in some natural Hilbert spaces...