Duke Mathematical Journal

Distortion elements in group actions on surfaces

John Franks and Michael Handel

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If G is a finitely generated group with generators {g1,,gj}, then an infinite-order element fG is a distortion element of G provided that liminfn|fn|/n=0, where |fn| is the word length of fn in the generators. Let S be a closed orientable surface, and let Diff(S)0 denote the identity component of the group of C1-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=T2 or S2. For μ a Borel probability measure on S, denote the group of C1-diffeomorphisms that preserve μ by Diffμ(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 Diffμ(S) with infinite image

Article information

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

First available in Project Euclid: 6 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37C85: Dynamics of group actions other than Z and R, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]
Secondary: 57M60: Group actions in low dimensions 22F10: Measurable group actions [See also 22D40, 28Dxx, 37Axx]


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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