2020/12/16 by Arnon Chor, Chor, Arnon · 1 citation
Mathematics · #37JXX (Secondary) #53DXX (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2012.08930
openalex publication_date 2020/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The group Ham(M,ω) of all Hamiltonian diffeomorphisms of a symplectic manifold (M,ω) 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,ω), in the case where M is a closed surface of genus 2 or 3. First, we prove that there exist diffeomorphisms in Ham(M,ω) arbitrarily far from being a k-th power, with respect to the metric, for any k ≥ 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,ω). 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.