Monotony of Certain Free Group Representations

Abstract

Let 1 be a free nonabelian group and let 0 be its boundary. Let ? h be one of the unitary representations of 1 introduced earlier by the authors in (1996, Duke Math. J. 82, 381 436). By its definition ? h acts on L 2 (0, d& h ) for a certain measure & h . This gives a boundary realization of ? h in a sense we make precise. We show that ? h does not have any other boundary realizations and simultaneously provide a new proof that ? h is irreducible. Academic Press

Asymptotic Schur orthogonality in hyperbolic groups with application to monotony

“…Here we will present an example of application of our results. We will deal with the phenomenon of monotony of representations, described by Kuhn and Steger in [20].…”

“…Let Γ be a non-abelian free group. By a generalized boundary representation (called boundary representation in [20]) of Γ on a Hilbert space H we will understand a pair (σ, ρ), where (1) σ is a unitary representation of Γ on H, (2) ρ is a representation of the C * -algebra C(∂Γ) on H, (3) σ and ρ satisfy the condition σ(g)ρ( f )σ(g −1 ) = ρ( f • g −1 ) for all g ∈ Γ and f ∈ C(∂Γ). In other words, this is a representation of the crossed product C(∂Γ) ⋊ Γ, where Γ acts on C(∂Γ) via g · f = f • g −1 .…”

“…As an application of the Asymptotic Orthogonality Theorem, we extend the results of Kuhn and Steger about monotony of free group representations. In [20], they propose a method of classifying the unitary representations of a free group Γ, which are weakly contained in the regular representation, using the notion of boundary realization. Roughly speaking, a boundary realization of a representation π on a Hilbert space H is a Γ-equivariant isometric embedding of H into a space of the form L 2 (∂Γ, µ), acted upon by the quasi-regular representation π µ corresponding to some measure µ, in such a way that the image is a cyclic subspace for the multiplication representation of C(∂Γ).…”

“…If a representation π of a free group is weakly contained in the regular representation, then it admits a boundary realization, and Kuhn and Steger distinguish three different behaviors of the class of boundary realizations of π, inspired by the case of restrictions of irreducible unitary representations of SL 2 (R) to a free lattice [20,Section 5]. They say that π is monotonous if it admits a unique boundary realization which is perfect, odd if it admits a unique boundary realization which is imperfect, and duplicitous if it admits exactly two perfect boundary realizations, and any imperfect realization is a combination of these two, in a certain precise sense.…”

Irreducible representations of the free product of groups

“…For those related to radial functions the method is to study the projection on a cyclic vector, see [FP1,FP2,PS,MZ,IP,Sz]. Interesting constructions of irreducible representations of the free group are due to Kuhn, Steger [KS3,KS4] and Paschke [P1, P2]. Let us also mention papers by M lotkowski [M2], Kuhn and Steger [KS2].…”