Eggbeater dynamics on symplectic surfaces of genus 2 and 3

TL;DR

This paper explores the geometry of Hamiltonian diffeomorphism groups on genus 2 and 3 surfaces, demonstrating the existence of elements far from being perfect powers and embedding free groups into their asymptotic cones.

Contribution

It extends previous results by showing the existence of diffeomorphisms far from k-th powers and embedding free groups into the asymptotic cone for genus 2 and 3 surfaces.

Findings

Existence of diffeomorphisms far from being k-th powers in $Ham(M,\omega)$

Embedding of the free group on two generators into the asymptotic cone

Generalization of previous work by Polterovich, Shelukhin, and Alvarez-Gavela

Abstract

The group $Ham(M,\omega)$ of all Hamiltonian diffeomorphisms of a symplectic manifold $(M,\omega)$ plays a central role in symplectic geometry. This group is endowed with the Hofer metric. In this paper we study two aspects of the geometry of $Ham(M,\omega)$, in the case where $M$ is a closed surface of genus 2 or 3. First, we prove that there exist diffeomorphisms in $Ham(M,\omega)$ arbitrarily far from being a $k$-th power, with respect to the metric, for any $k \geq 2$. This part generalizes previous work by Polterovich and Shelukhin. Second, we show that the free group on two generators embeds into the asymptotic cone of $Ham(M,\omega)$. This part extends previous work by Alvarez-Gavela et al. Both extensions are based on two results from geometric group theory regarding incompressibility of surface embeddings.