2019/07/03 by Nancy Guelman, Guelman, Nancy, Isabelle Liousse +1
Computer Science · Mathematics · #Cellular Automata and Applications #Computability, Logic, AI Algorithms #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1907.01808
openalex publication_date 2019/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Gn ≃ (\mathbb S1)n \rtimes\mathcal Sn , where \mathbb S1 is the circle and \mathcal Sn 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 Gn 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 Z2. 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.