2004/04/29 by John Franks, Michael Handel, Franks, John +1 · 1 citation
Mathematics · #37E30 #57S25 #Advanced Algebra and Geometry #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #math.DS #msc:37E30 #msc:57S25
paper · pdf · doi:10.48550/arxiv.math/0404532
openalex publication_date 2004/04/29 · arxiv created 2005/03/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If \G is a finitely generated group with generators \g1,...,gj\ then an infinite order element f ∈ \G is a \em distortion element of \G provided \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 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.