Random Heegaard splittings

2008/09/30 by Joseph Maher · 1 citation
Mathematics · #Class (philosophy) #Combinatorics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Harmonic #Heegaard splitting #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Random walk #Statistics #math.GT #msc:20F65 #msc:37E30 #msc:57M50

paper · pdf · doi:10.1112/jtopol/jtq031

31 pages, 5 figures, revised version

openalex publication_date 2010/01/01 · arxiv created 2010/11/16 · arxiv updated 2014/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Consider a random walk on the mapping class group, and let wn be the location of the random walk at time n. A random Heegaard splitting M(wn) is a 3-manifold obtained by using wn as the gluing map between two handlebodies. We show that the joint distribution of (wn, wn−1) is asymptotically independent, and converges to the product of the harmonic and reflected harmonic measures defined by the random walk. We use this to show that the translation length of wn acting on the curve complex, and the distance between the disk sets of M(wn) in the curve complex, grows linearly in n. In particular, this implies that a random Heegaard splitting is hyperbolic with asymptotic probability 1.

Citations

• 3-Manifolds as viewed from the curve complex
• Three dimensional manifolds, Kleinian groups and hyperbolic geometry
• Metric Spaces of Non-Positive Curvature
• Ricci flow with surgery on three-manifolds

Cited by

• Random walks on the mapping class group
• Hyperbolic 3-manifold [wikipedia]

Related

• Uniform models and short curves for random 3-manifolds
• Statistics of Random 3-Manifolds occasionally fibering over the circle
• Random methods in 3-manifold theory
• Effective hyperbolization and length bounds for Heegaard splittings
• Random walks on hyperbolic groups and their Riemann surfaces