2007/11/30 by Martin Lustig, Yoav Moriah · 1 citation
Mathematics · #Combinatorics #Computer science #Countable set #Dehn surgery #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Lebesgue integration #Lebesgue measure #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Mathematics #Measure (data warehouse) #Pure mathematics #Quotient #Set (abstract data type) #Simple (philosophy) #Space (punctuation) #Surface (topology) #math.GT #msc:57M99
paper · pdf · doi:10.1112/jtopol/jtq022
This version contains a substantially stronger version of the main theorem. 31 pages 4 figures
arxiv created 2009/04/14 · openalex publication_date 2010/01/01 · arxiv updated 2014/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We introduce a general notion of ‘genericity’ for countable subsets of a space with Borel measure, and apply it to the set of vertices in the curve complex of a surface Σ, interpreted as subset of the space of projective measured laminations in Σ, equipped with its natural Lebesgue measure. We prove that, for any 3-manifold M, the set of curves c on a Heegaard surface Σ ⊂ M, such that every non-trivial Dehn twist at c yields a Heegaard splitting of high distance, is generic in the set of all essential simple closed curves on Σ. Our definition of ‘genericity’ is different and more intrinsic than alternative such existing notions, given, for example, via random walks or via limits of quotients of finite sets.