Balanced curves, quasimorphisms, and the equator conjecture

TL;DR

This paper introduces new homogeneous quasimorphisms on the Hamiltonian diffeomorphism group of the sphere and proves an infinite diameter property for the space of equators under certain metrics, using novel tools inspired by curve graph theory.

Contribution

It constructs an infinite family of homogeneous quasimorphisms and proves an analogue of the equator conjecture with new methods inspired by curve graph theory.

Findings

Existence of infinite-dimensional family of homogeneous quasimorphisms

Unbounded quasimorphisms vanishing on small-area supports

Infinite diameter of equator space under fragmentation metrics

Abstract

We construct a new infinite-dimensional family of homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-sphere. Moreover, for any constant $K$ less than the total area of the sphere, we produce unbounded homogeneous quasimorphisms that vanish on any map supported on some disk of area at most $K$. As an application, we prove an analogue of the equator conjecture, namely that the space of equators equipped with any choice of quantitative fragmentation metric has infinite diameter. To prove our results, we introduce tools that draw inspiration from the theory of curve graphs used to study mapping class groups.