## Abstract

If G is a finitely generated group with generators [g_{1}, . . . , g_{j}}, then an infinite-order element f ∈ G is a distortion element of G provided that lim inf _{n→ ∞} |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 that if f is a distortion element in some finitely generated subgroup of Diff(S)_{0}, then supp(μ) ⊂ Fix(f) for every f-invariant Borel probability measure μ. Related results are proved for S = T^{2} or S^{2}. For μ a Borel probability measure on S, denote the group of C^{1}- diffeomorphisms that preserve μ by Diff_{μ}(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_{μ}(S) with infinite image.

Original language | English (US) |
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Pages (from-to) | 441-468 |

Number of pages | 28 |

Journal | Duke Mathematical Journal |

Volume | 131 |

Issue number | 3 |

DOIs | |

State | Published - Feb 15 2006 |

## ASJC Scopus subject areas

- Mathematics(all)