Distortion elements in group actions on surfaces

Abstract

If $\mathcal{G}$ is a finitely generated group with generators $\{g_1,\dots ,g_j\}$, then an infinite-order element $f\in \mathcal{G}$ is a distortion element of $\mathcal{G}$ provided that $lim\hspace{1em}{inf}_{n\to \infty }\left|f^n\right|/n=0,$ where $\left|f^n\right|$ is the word length of $f^n$ in the generators. Let $S$ be a closed orientable surface, and let $\mathrm{Diff}(S){}_0$ 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 that if $f$ is a distortion element in some finitely generated subgroup of $\mathrm{Diff}(S){}_0$, then $\mathrm{supp}(\mu )\subset \mathrm{Fix}(f)$ 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 ${\mathrm{Diff}}_{\mu }(S)$. We give several applications of our main result, showing that certain groups, including a large class of higher-rank lattices, admit no homomorphisms to ${\mathrm{Diff}}_{\mu }(S)$ with infinite image