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.