  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.
