Relations with a fixed interval exchange transformation

We study the group of all interval exchange transformations (IETs). Katok asked whether it contains a free subgroup. We show that for every IET S, there exists a dense open set Ω(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega (S)$$\end{document} of admissible IETs such that the group generated by S and any T∈Ω(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in \Omega (S)$$\end{document} is not free of rank 2. This extends a result by Dahmani et al. (Groups Geom Dyn 7(4):883–910, 2013): the group generated by a generic pair of elements of IET([0;1)) is not free (assuming a suitable condition on the underlying permutation).