Distortion elements in group actions on surfaces

If G is a finitely generated group with generators [g₁, . . . , g_j}, 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 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.