  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
$k \geq 2$. 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.
