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Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire

Juliette Bavard

Geometry & Topology 20 (2016) 491–535

Le groupe modulaire Γ du plan privé d’un ensemble de Cantor apparaît naturellement en dynamique. On montre ici que le graphe des rayons, analogue du complexe des courbes pour cette surface de type infini, est de diamètre infini et hyperbolique. On utilise l’action de Γ sur ce graphe hyperbolique pour exhiber un quasi-morphisme non trivial explicite sur Γ et pour montrer que le deuxième groupe de cohomologie bornée de Γ est de dimension infinie. On donne enfin un exemple d’un élément hyperbolique de Γ dont la longueur stable des commutateurs est nulle. Ceci réalise un programme proposé par Danny Calegari.

The mapping class group Γ of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of Γ on this graph to find an explicit non trivial quasimorphism on Γ and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a hyperbolic element of Γ with vanishing stable commutator length. This carries out a program proposed by Danny Calegari.

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mapping class groups, surface homeomorphisms, quasimorphisms, Gromov-hyperbolic space, Cantor sets
Mathematical Subject Classification 2010
Primary: 37E30
Secondary: 57M60, 20F65
Received: 23 September 2014
Accepted: 21 May 2015
Published: 29 February 2016
Proposed: Danny Calegari
Seconded: Leonid Polterovich, Bruce Kleiner
Juliette Bavard
Université Pierre et Marie Curie
4 place Jussieu
75252 Paris Cedex 05