Distortion for diffeomorphisms of surfaces with boundary

If $G$ is a finitely generated group with generators $\{g_1,..., g_s\}$, we say an infinite-order element $f \in G$ is a distortion element of $G$ provided that $\displaystyle \liminf_{n \to \infty} \frac{|f^n|}{n} = 0$, where $|f^n|$ is the word length of $f^n$ with respect to the given generators. Let $S$ be a compact orientable surface, possibly with boundary, and let $\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result is that if $S$ has genus at least two, and $f$ is a distortion element in some finitely generated subgroup of $\Diff(S)_0$, then $\supp(\mu) \subseteq \Fix(f)$ for every $f$-invariant Borel probability measure $\mu$. Under a small additional hypothesis the same holds in lower genus. For $\mu$ a Borel probability measure on $S$, denote the group of $C^1$ diffeomorphisms that preserve $\mu$ by $\Diff_\mu(S)$. Our main result implies that a large class of higher-rank lattices admit no homomorphisms to $\Diff_{\mu}(S)$ with infinite image. These results generalize those of Franks and Handel to surfaces with boundary.