Hyperbolicit\'e du graphe des rayons et quasi-morphismes sur un gros groupe modulaire

TL;DR

This paper proves the hyperbolicity and infinite diameter of the ray graph associated with a certain infinite-type surface, constructs explicit quasimorphisms on the mapping class group, and explores properties of hyperbolic elements.

Contribution

It establishes the hyperbolic nature of the ray graph for the mapping class group of a Cantor set complement and constructs explicit quasimorphisms, advancing understanding of the group's bounded cohomology.

Findings

Ray graph has infinite diameter and is hyperbolic.

Constructed explicit non-trivial quasimorphisms on the group.

Identified hyperbolic elements with vanishing stable commutator length.

Abstract

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.