Abelianization of some groups of interval exchanges
Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 59-108.

Let IET be the group of permutations of [0,1[ which are continuous outside a finite set, right-continuous and piecewise translations. The abelianization homomorphism of IET was described by Arnoux–Fathi and Sah. It gives an isomorphism between the abelianization of IET and the second exterior power of the reals over the rationals.

Let Γ be a subgroup of / and let Γ ˜ be its preimage in . We denote by IET(Γ) the subgroup of IET consisting of all elements continuous outside Γ. We establish an explicit isomorphism between its abelianization and the second skew-symmetric power of Γ ˜ over . This group often has non-trivial 2-torsion.

Then, we define IET as the group of all interval exchange transformations with flips. For every subgroup IET (Γ) we establish an explicit isomorphism between its abelianization and an explicit 2-elementary abelian group.

Soit IET le groupe des permutations de [0,1[ qui sont continues en dehors d’un ensemble fini, continues à droite et qui sont des translations par morceaux. Arnoux–Fathi et Sah ont établi un isomorphisme entre l’abélianisé d’IET et la seconde puissance extérieure des réels sur les rationnels.

Soit Γ un sous-groupe de / et Γ ˜ sa pré-image. On note par IET(Γ) le sous-groupe d’IET composé de l’ensemble des éléments continus en dehors de Γ. On établit un isomorphisme explicite entre l’abélianisé d’IET(Γ) et la seconde puissance antisymétrique de Γ ˜ sur . Ce groupe a souvent de la 2-torsion.

Puis nous définissons IET comme le groupe de tous les échanges d’intervalles avec renversements. Pour tout sous-groupe IET (Γ), on établit un isomorphisme explicite entre son abélianisé et un groupe abélien 2-élémentaire explicite.

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DOI: 10.5802/aif.3466
Classification: 37E05,  20F65,  20J06
Keywords: Interval exchange, abelianization.
Lacourte, Octave 1

1 University Claude Bernard Lyon 1 Institut Camille Jordan 43 blvd du 11 novembre 1918 69622 Villeurbanne, France
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Lacourte, Octave. Abelianization of some groups of interval exchanges. Annales de l'Institut Fourier, Volume 72 (2022) no. 1, pp. 59-108. doi : 10.5802/aif.3466. https://aif.centre-mersenne.org/articles/10.5802/aif.3466/

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