[Paper Review] Gromov-Hyperbolicity of the ray graph and quasimorphisms on a big mapping class group

Juliette Bavard|arXiv (Cornell University)|Feb 8, 2018

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper establishes that the ray graph for the plane minus a Cantor set is Gromov-hyperbolic and has infinite diameter, enabling the construction of a nontrivial quasimorphism on the big mapping class group Γ. Using this action on a hyperbolic space, the authors prove that Γ has infinite-dimensional second bounded cohomology and exhibit a hyperbolic element with vanishing stable commutator length, fulfilling a program by Danny Calegari.

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.

Motivation & Objective

• To study the big mapping class group Γ of the plane minus a Cantor set, which arises naturally in dynamics through group actions on the plane with bounded orbits.
• To understand the large-scale geometry of Γ by analyzing its action on the ray graph, a hyperbolic analog of the curve complex for infinite-type surfaces.
• To construct nontrivial quasimorphisms on Γ and show that its second bounded cohomology is infinite-dimensional.
• To find a hyperbolic element in Γ with vanishing stable commutator length, addressing a question posed by Calegari.

Proposed method

• Define the ray graph as the simplicial complex whose vertices are isotopy classes of proper rays from a Cantor set point to infinity, with edges between disjoint rays.
• Prove the ray graph has infinite diameter by constructing a sequence of rays whose distances grow without bound.
• Establish Gromov-hyperbolicity of the ray graph using geometric and inductive arguments on ray sequences.
• Construct a loxodromic element g in Γ that acts by translation on a geodesic ray, using inductive ray constructions with alternating clockwise and counterclockwise turns.
• Use the action of Γ on the hyperbolic ray graph to define a nontrivial quasimorphism via the translation length on the geodesic axis.
• Leverage the existence of nontrivial quasimorphisms to show that the second bounded cohomology of Γ is infinite-dimensional.

Experimental results

Research questions

• RQ1 Is the ray graph for the plane minus a Cantor set hyperbolic in the sense of Gromov?
• RQ2 Does the ray graph have infinite diameter, indicating non-trivial large-scale geometry for the big mapping class group?
• RQ3 Can the action of the big mapping class group on the ray graph be used to construct nontrivial quasimorphisms?
• RQ4 Does the group Γ admit a hyperbolic element with vanishing stable commutator length?
• RQ5 What is the dimension of the second bounded cohomology group of Γ?

Key findings

• The ray graph has infinite diameter, demonstrating that the large-scale geometry of the big mapping class group is nontrivial and complex.
• The ray graph is Gromov-hyperbolic, providing a hyperbolic space on which the big mapping class group acts nontrivially.
• There exists a nontrivial quasimorphism on the big mapping class group Γ, constructed via the translation length on a geodesic axis in the ray graph.
• The second bounded cohomology group of Γ is infinite-dimensional, as shown through the existence of infinitely many linearly independent quasimorphisms.
• A hyperbolic element h ∈ Γ exists with vanishing stable commutator length, answering a question posed by Calegari.
• The group Γ acts loxodromically on the ray graph, with a specific element g acting by translation on a geodesic sequence (γ_k), satisfying g^n(γ_0) = γ_{2n} for all n ∈ ℕ.