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 Omega(S) of admissible IETs such that the group generated by S and any T is an element of Omega(S) 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).