If G is a finitely generated group with generators [g1, . . . , gj}, then an infinite-order element f ∈ G is a distortion element of G provided that lim inf n→ ∞ |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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