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

By Juliette Bavard. 2014

51 citations recorded by the source corpus.

These notes are the English version of the paper "Hyperbolicit\'e 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/713c3136c74f47624274da3acf066df28163986b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Large-scale geometry of big mapping class groups (2019, background): “9 also recovers, with a new proof, some of the work of Bavard in [3] and Durham-Fanoni-Vlamis in [7].”
Algebraic and topological properties of big mapping class groups (2017, methodology): “ been a recent trend aimed at understanding inﬁnite-type surfaces and their mapping class groups. For instance, recent work has investigated big mapping class group actions on hyperbolic graphs (e.g. [7,3,19]). This article focuses on the algebraic structure of big mapping class groups. Our methods ”
Quasi-morphisms on surface diffeomorphism groups (2019, methodology): “Calegari, variants of curve graphs have also been dened and used to construct quasi-morphisms on so-called big mapping class groups, e.g. the mapping class group of the plane minus a Cantor set (see [Bav16]). Intuitively, curve graphs encode intersection patterns of isotopy classes of curves (or sim”
The First Integral Cohomology of Pure Mapping Class Groups (2017, background): “The work to date has largely focused on highlighting the similarities between infinite-type mapping class groups and their finite-type counterparts [1, 4, 5, 10, 14, 22].”