Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group

Abstract

These notes are the English version of the paper "Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma 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 Gamma on this graph to find an explicit non trivial quasimorphism on Gamma and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a hyperbolic element of Gamma with vanishing stable commutator length. This carries out a program proposed by Danny Calegari.

arXiv comment34 pages, 25 figures, comments are welcome

References in corpus (3)

• Geometry of the Complex of Curves I: Hyperbolicity
• Bounded cohomology of subgroups of mapping class groups
• Bounded cohomology and non-uniform perfection of mapping class groups

Cited by in corpus (8)

• Big mapping class groups: an overview
• Large scale geometry of big mapping class groups
• The first integral cohomology of pure mapping class groups
• Algebraic and topological properties of big mapping class groups
• Big mapping class groups and rigidity of the simple circle
• Loxodromic elements in big mapping class groups via the Hooper-Thurston-Veech construction
• Braided Thompson groups with and without quasimorphisms
• Infinite-type loxodromic isometries of the relative arc graph