# Quasi-isometry type of the metric space derived from the kernel of the Calabi homomorphism

## TL;DR

This paper demonstrates that the metric space formed from the kernel of the Calabi homomorphism on area-preserving diffeomorphisms of the 2-disk has a complex geometric structure, not resembling a simple half-line.

## Contribution

It establishes the quasi-isometry type of the metric space derived from the kernel of the Calabi homomorphism, revealing its non-linear geometric nature.

## Key findings

- The space is not quasi-isometric to the half line.
- The structure of the kernel's conjugacy classes is complex.
- The result provides insight into the geometry of symplectic diffeomorphism groups.

## Abstract

We prove that the set of symmetrized conjugacy classes of the kernel of the Calabi homomorphism on the group of area-preserving diffeomorphisms of the $2$-disk is not quasi-isometric to the half line.