An ergodic theorem for the quasi-regular representation of the free group

Abstract

In [BM11], an ergodic theoremà la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized [Boy15] to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of [Rob03], with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group Fr of rank r even if Fr is not the fundamental group of a closed manifold, and may have an arithmetic spectrum. X u := {x ∈ X | u is a prefix of ⌊x⌋} B u := {ξ ∈ B | u is a prefix of ξ}

Radial rapid decay does not imply rapid decay

“…The uniform boundedness condition has been studied in [BM11], [Boy16], [Gar14], [BLP17] and [BPL17] where authors investigate generalizations of the von Neumann ergodic theorem to the situation where the measure is only quasi-invariant. In several cases of interest, the relevant generalization of von Neumann means are the normalized means 1…”

Free group representations from vector-valued multiplicative functions, III

“…In [13] it is proved that any irreducible matrix system admits, up to a normalization, a unique (up to scalar multiples) tuple of strictly positive definite forms (B a ) satisfying (3).…”

Ergodic boundary representations

“…In the case the CAT(−1) space is the Cayley graph (with all edges of length 1) of the free group on n letters, all closed geodesics in the wedge of n circles of length 1 have integral length, hence the spectrum is obviously arithmetic and the general theory does not apply. Nevertheless a direct counting argument [9] allows to prove Bader-Muchnik's theorem in this case as well. Following the same lines of ideas as Bader and Muchnik, L. Garncarek was able to prove that the boundary representation associated to a Patterson-Sullivan measure of a Gromov hyperbolic group is irreducible [16].…”

“…4.1.1]. For free groups, a direct counting gives the result [9] and it would be interesting to understand how the normalizing constant in [27, Thm. 4.1.1] varies with α > 0 in the case of the tree covering the wedge of two circles of lengths 1 and α.…”