Centralizers in the Group of Interval Exchange Transformations

Daniel Bernazzani

Published 2016-08-25Version 1

Let $T$ be an $m$-interval exchange transformation. By the rank of $T$ we mean the dimension of the $\mathbb{Q}$-vector space spanned by the lengths of the exchanged subintervals. We prove that if $T$ satisfies Keane's infinite distinct orbit condition and $\text{rank}(T)>1+\lfloor m/2 \rfloor$ then the only interval exchange transformations which commute with $T$ are its powers. The main step in our proof is to show that the centralizer of $T$ is torsion-free under the above hypotheses.