Distortion for diffeomorphisms of surfaces with boundary

TL;DR

This paper investigates distortion elements in groups of surface diffeomorphisms, showing that such elements have fixed support on surfaces of genus at least two, with implications for homomorphisms from higher-rank lattices.

Contribution

It extends the understanding of distortion elements in surface diffeomorphism groups to surfaces with boundary and higher genus, generalizing previous results by Franks and Handel.

Findings

Distortion elements have support contained in their fixed point set.

Higher-rank lattices cannot have infinite image homomorphisms into certain diffeomorphism groups.

Results apply to surfaces with boundary and lower genus under additional hypotheses.

Abstract

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.