Commensurating actions for groups of piecewise continuous transformations

Cornulier, Yves

Annales Henri Lebesgue, Tome 4 (2021), pp. 1457-1520.

Nous utilisons le formalisme des actions partielles, dû à Exel, pour construire différentes actions commensurantes. Nous nous en servons pour des groupes préservant, par morceaux, une structure géométrique, et nous interprétons la propriété transfixante de ces actions commensurantes comme l’existence d’un modèle pour lequel le groupe agit en préservant la structure géométrique. Cela s’applique à divers groupes par morceaux en dimension 1, notamment de classe ${𝒞}^k$ par morceaux, affines par morceaux, projectifs par morceaux.

On en déduit des résultats de conjugaison pour les sous-groupes avec la propriété FW, ou pour les sous-groupes cycliques distordus. Par exemple on obtient, sous des hypothèses convenables, la conjugaison d’une action affine par morceaux à une action affine, sur un autre modèle. Avec la même méthode, on obtient un résultat similaire dans le cas projectif. À titre d’illustration, un corollaire est le fait que le groupe des transformations projectives par morceaux du cercle n’a pas de sous-groupe infini avec la propriété T de Kazhdan  ; ce corollaire est nouveau même dans le cas affine par morceaux.

De plus, on obtient avec cela la classification des copies topologiques du groupe des rotations du cercle dans le groupe des homéomorphismes projectifs par morceaux du cercle. Le cas affine par morceaux est un résultat classique de Minakawa.

We use partial actions, as formalized by Exel, to construct various commensurating actions. We use this in the context of groups piecewise preserving a geometric structure, and we interpret the transfixing property of these commensurating actions as the existence of a model for which the group acts preserving the geometric structure. We apply this to many piecewise groups in dimension 1, notably piecewise of class ${𝒞}^k$, piecewise affine, piecewise projective (possibly discontinuous).

We derive various conjugacy results for subgroups with Property FW, or distorted cyclic subgroups. For instance we obtain, under suitable assumptions, the conjugacy of a given piecewise affine action to an affine action on possibly another model. By the same method, we obtain a similar result in the projective case. An illustrating corollary is the fact that the group of piecewise projective self-transformations of the circle has no infinite subgroup with Kazhdan’s Property T; this corollary is new even in the piecewise affine case.

In addition, we use this to provide the classification of circle subgroups of piecewise projective homeomorphisms of the projective line. The piecewise affine case is a classical result of Minakawa.

Classification : 57S05, 57M50, 57M60, 18B40, 20F65, 22F05, 37B99, 53C10, 53C15, 53C29, 57R30, 57S25, 57S30, 58H05
Mots clés : Piecewise linear transformations, Thompson groups, commensurating actions, partial actions, Property FW, geometric structures, affine 1-manifolds, projective 1-manifolds

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