Eggbeater dynamics on symplectic surfaces of genus 2 and 3

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 $k\ge 2$. 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.