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15 February 2006 Distortion elements in group actions on surfaces
John Franks, Michael Handel
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Duke Math. J. 131(3): 441-468 (15 February 2006). DOI: 10.1215/S0012-7094-06-13132-0

Abstract

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

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John Franks. Michael Handel. "Distortion elements in group actions on surfaces." Duke Math. J. 131 (3) 441 - 468, 15 February 2006. https://doi.org/10.1215/S0012-7094-06-13132-0

Information

Published: 15 February 2006
First available in Project Euclid: 6 February 2006

zbMATH: 1088.37009
MathSciNet: MR2219247
Digital Object Identifier: 10.1215/S0012-7094-06-13132-0

Subjects:
Primary: 37C85
Secondary: 22F10, 57M60

Rights: Copyright © 2006 Duke University Press

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Vol.131 • No. 3 • 15 February 2006
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