On the autonomous metric on groups of Hamiltonian diffeomorphisms of closed hyperbolic surfaces

TL;DR

This paper proves that the group of Hamiltonian diffeomorphisms on a closed hyperbolic surface is unbounded when measured with the autonomous metric, highlighting its rich geometric structure.

Contribution

It establishes the unboundedness of the autonomous metric on the Hamiltonian diffeomorphism group of closed hyperbolic surfaces, a previously unknown property.

Findings

The autonomous metric is bi-invariant.

The group $Ham( ext{surface})$ is unbounded under this metric.

Provides new insights into the geometric properties of Hamiltonian diffeomorphism groups.

Abstract

Let $\Sigma_g$ be a closed hyperbolic surface of genus $g$ and let $Ham(\Sigma_g)$ be the group of Hamiltonian diffeomorphisms of $\Sigma_g$. The most natural word metric on this group is the autonomous metric. It has many interesting properties, most important of which is the bi-invariance of this metric. In this work we show that $Ham(\Sigma_g)$ is unbounded with respect to this metric.