Distortion Elements in Group actions on surfaces

If $\text{G}$ is a finitely generated group with generators $\{g_1,\mathrm{.}\mathrm{.}\mathrm{.},g_j\}$ then an infinite order element $f\in \text{G}$ is a {\em distortion element} of $\text{G}$ provided ${\displaystyle \underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\mathrm{\mid }{f}^n\mathrm{\mid }\mathrm{/}n=0,}$ where $\mathrm{\mid }{f}^n\mathrm{\mid }$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface and let $\text{Diff}(S{)}_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result shows that if $S$ has genus at least two and if $f$ is a distortion element in some finitely generated subgroup of $\text{Diff}(S{)}_0$, then $\text{supp}(\mu )\subset \text{Fix}(f)$ for every $f$-invariant Borel probability measure $\mu$. Related results are proved for $S=T^2$ or $S^2$. For $\mu$ a Borel probability measure on $S$, denote the group of $C^1$ diffeomorphisms that preserve $\mu$ by ${\text{Diff}}_{\mu }(S)$. We give several applications of our main result showing that certain groups, including a large class of higher rank lattices, admit no homomorphisms to ${\text{Diff}}_{\mu }(S)$ with infinite image.