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

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.