## Duke Mathematical Journal

### Distortion elements in group actions on surfaces

#### Abstract

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

#### Article information

Source
Duke Math. J., Volume 131, Number 3 (2006), 441-468.

Dates
First available in Project Euclid: 6 February 2006

https://projecteuclid.org/euclid.dmj/1139232346

Digital Object Identifier
doi:10.1215/S0012-7094-06-13132-0

Mathematical Reviews number (MathSciNet)
MR2219247

Zentralblatt MATH identifier
1088.37009

#### Citation

Franks, John; Handel, Michael. Distortion elements in group actions on surfaces. Duke Math. J. 131 (2006), no. 3, 441--468. doi:10.1215/S0012-7094-06-13132-0. https://projecteuclid.org/euclid.dmj/1139232346

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