Subgroup of interval exchanges generated by torsion elements and rotations

2012/08/05 by Boshernitzan, Michael
#20Exx #37E05 #54H15 #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1208.1023

Abstract

Denote by G the group of interval exchange transformations (IETs) on the unit interval. Let G_per⊂ G be the subgroup generated by torsion elements in G (periodic IETs), and let G_rot⊂ G be the subset of 2-IETs (rotations). The elements of the subgroup G₁=< G_per,G_rot>⊂ G (generated by the sets G_per and G_rot) 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\subsetneq G. The main tools used in the paper are the SAF invariant and a recent result by Y. Vorobets that G_per coincides with the commutator subgroup of G.

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