2012/02/16 by Parkhe, Kiran
#22F10 #37C85 #57M60 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1202.3536
If G is a finitely generated group with generators \g1,..., gs\, we say an infinite-order element f ∈ G is a distortion element of G provided that \liminfn → ∞ (|fn|)/(n) = 0, where |fn| is the word length of fn with respect to the given generators. Let S be a compact orientable surface, possibly with boundary, and let \Diff(S)0 denote the identity component of the group of C1 diffeomorphisms of S. Our main result is that if S has genus at least two, and 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. Under a small additional hypothesis the same holds in lower genus. For μ a Borel probability measure on S, denote the group of C1 diffeomorphisms that preserve μ by \Diffμ(S). Our main result implies that a large class of higher-rank lattices admit no homomorphisms to \Diffμ(S) with infinite image. These results generalize those of Franks and Handel to surfaces with boundary.