Stability in orbit equivalence for Baumslag-Solitar groups and Vaes groups

A measure-preserving action of a discrete countable group on a standard probability space is called stable if the associated equivalence relation is isomorphic to its direct product with the ergodic hyperfinite equivalence relation of type II_1. We show that any Baumslag-Solitar group has such an ergodic, free and stable action. It follows that any Baumslag-Solitar group is measure equivalent to its direct product with any amenable group. The same property is obtained for the inner amenable groups of Vaes.Comment: 23 pages, references added and updated (v2), minor changes (v3

Quotients and subgroups of Baumslag-Solitar groups

We determine all generalized Baumslag-Solitar groups (finitely generated groups acting on a tree with all stabilizers infinite cyclic) which are quotients of a given Baumslag-Solitar group BS(m,n), and (when BS(m,n) is not Hopfian) which of them also admit BS(m,n) as a quotient. We determine for which values of r,s one may embed BS(r,s) into a given BS(m,n), and we characterize finitely generated groups which embed into some BS(n,n).Comment: Final version, to appear in Journal of Group Theor

A note on the von Neumann algebra of a Baumslag-Solitar group

We study qualitative properties of the group von Neumann algebra of a Baumslag-Solitar group. Namely, we prove that, in the non-amenable and {ICC} case, the associated ${\rm II}_1$ factor is prime, not solid, and does not have any Cartan subalgebra

Orthonormal dilations of Parseval wavelets

We prove that any Parseval wavelet frame is the projection of an orthonormal wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=< u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in some special cases, and show that for wavelet sets, it is related to symbolic dynamics. We show that the structure of the representation depends on the analysis of certain finite orbits for the associated symbolic dynamics. We give concrete examples of Parseval wavelets for which we compute the orthonormal dilations in detail; we show that there are examples of Parseval wavelet sets which have infinitely many non-isomorphic orthonormal dilations.Comment: v2, improved introduction according to the referee's suggestions, corrected some typos. Accepted for Mathematische Annale

A context-free and a 1-counter geodesic language for a Baumslag-Solitar group

We give a language of unique geodesic normal forms for the Baumslag-Solitar group BS(1,2) that is context-free and 1-counter. We discuss the classes of context-free, 1-counter and counter languages, and explain how they are inter-related