Reversible Maps and Products of Involutions in Groups of IETS

Abstract

An element $f$ of a group $G$ is reversible if it is conjugated in $G$ to its own inverse; when the conjugating map is an involution, $f$ is called strongly reversible. We describe reversible maps in certain groups of interval exchange transformations namely $G_n\simeq ({\mathbb{S}}^1{)}^n⋊{\mathcal{S}}_n$, where ${\mathbb{S}}^1$ is the circle and ${\mathcal{S}}_n$ is the group of permutations of $\{1,...,n\}$. We first characterize strongly reversible maps, then we show that reversible elements are strongly reversible. As a corollary, we obtain that composites of involutions in $G_n$ are product of at most four involutions. We prove that any reversible Interval Exchange Transformation (IET) is reversible by a finite order element and then it is the product of two periodic IETs. In the course of proving this statement, we classify the free actions of $BS(1,-1)$ by IET and we extend this classification to free actions of finitely generated torsion free groups containing a copy of ${\mathbb{Z}}^2$. We also give examples of faithful free actions of $BS(1,-1)$ and other groups containing reversible IETs. We show that periodic IETs are product of at most $2$ involutions. For IETs that are products of involutions, we show that such 3-IETs are periodic and then are product of at most $2$ involutions and we exhibit a family of non periodic 4-IETs for which we prove that this number is at least $3$ and at most $6$.