  If $\G$ is a finitely generated group with generators $\{g_1,...,g_j\}$ then
an infinite order element $f \in \G$ is a {\em distortion element} of $\G$
provided $\displaystyle{\liminf_{n \to \infty} |f^n|/n = 0,}$ where $|f^n|$ is
the word length of $f^n$ in the generators. Let $S$ be a closed orientable
surface and let $\Diff(S)_0$ 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 $\Diff(S)_0$, then $\supp(\mu) \subset \Fix(f)$ 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 $\Diff_{\mu}(S)$. We give several
applications of our main result showing that certain groups, including a large
class of higher rank lattices, admit no homomorphisms to $\Diff_{\mu}(S)$ with
infinite image.
