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

Rigidity of amalgamated free products in measure equivalence

A discrete countable group \Gamma is said to be ME rigid if any discrete countable group that is measure equivalent to \Gamma is virtually isomorphic to \Gamma. In this paper, we construct ME rigid groups by amalgamating two groups satisfying rigidity in a sense of measure equivalence. A class of amalgamated free products is introduced, and discrete countable groups which are measure equivalent to a group in that class are investigated.Comment: 51 pages, Theorem 1.5 (a) and Proposition 9.5 modifie

The quasi-periodic doubling cascade in the transition to weak turbulence

The quasi-periodic doubling cascade is shown to occur in the transition from regular to weakly turbulent behaviour in simulations of incompressible Navier-Stokes flow on a three-periodic domain. Special symmetries are imposed on the flow field in order to reduce the computational effort. Thus we can apply tools from dynamical systems theory such as continuation of periodic orbits and computation of Lyapunov exponents. We propose a model ODE for the quasi-period doubling cascade which, in a limit of a perturbation parameter to zero, avoids resonance related problems. The cascade we observe in the simulations is then compared to the perturbed case, in which resonances complicate the bifurcation scenario. In particular, we compare the frequency spectrum and the Lyapunov exponents. The perturbed model ODE is shown to be in good agreement with the simulations of weak turbulence. The scaling of the observed cascade is shown to resemble the unperturbed case, which is directly related to the well known doubling cascade of periodic orbits

Commensurators of surface braid groups

We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.Comment: 35 pages, 12 figure