Freiman's theorem in an arbitrary nilpotent group
  We prove a Freiman-Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate subgroup A of an s-step nilpotent group G is contained in a coset nilprogression of rank at most f(K) and cardinality at most exp(g(K))|A|, with f and g polynomials depending only on the step s of G. To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal'cev's embedding theorem. 
Matthew Tointon
math.CO math.GR