Subgroup of interval exchanges generated by torsion elements and rotations

Abstract

Denote by G the group of interval exchange transformations (IETs) on the unit interval. Let Gper ⊂ G be the subgroup generated by torsion elements in G (periodic IETs), and let Grot ⊂ G be the subset of 2-IETs (rotations). The elements of the subgroup G1 = hGper,Groti ⊂ G (generated by the sets Gper and Grot) are characterized constructively in terms of their Sah-Arnoux-Fathi (SAF) invariant. The characterization implies that a non-rotation type 3-IET lies in G1 if and only if the lengths of its exchanged intervals are linearly dependent over Q. In particular, G1 ( G. The main tools used in the paper are the SAF invariant and a recent result by Y. Vorobets that Gper coincides with the commutator subgroup of G. 1. A group of IETs Denote by R, Q, N the sets of real, rational and natural numbers. By a standard interval we mean a finite interval of the form X = (a,b) ⊂ R (left closed - right open). We write |X| = b − a for its length. By an IET (interval exchange transformation) we mean a pair (X,f) where X = (a,b) is a standard interval and f is a right continuous bijection f : X → X with a finite set D of discontinuities and such that the translation function γ(x) = f(x)−x is piecewise constant. The map f itself is often referred to as IET, and then X = domain(f) and D = disc(f) denote the domain (also the range) of f and the discontinuity set of f, respectively. Given an IET f : X → X, the set disc(f) partitions X into a finite number of subintervals Xk in such a way that f restricted to each Xk is a translation