Paper · Oct 21, 2019

Uniform models and short curves for random 3-manifolds

Peter Feller, Alessandro Sisto, Gabriele Viaggi

math.GTarXiv:1910.09486v3
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Abstract

We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1. These constructions provide a lot of control on the resulting metric, allowing us to prove various results about the coarse growth rate of geometric invariants, such as diameter and injectivity radius, and about arithmeticity and commensurability in families of random 3-manifolds. For example, we show that the diameter of a random Heegaard splitting grows coarsely linearly in the length of the associated random walk. The constructions only use tools from the deformation theory of Kleinian groups, that is, we do not rely on the solution of the Geometrization Conjecture by Perelman. In particular, we give a proof of Maher's result that random 3-manifolds are hyperbolic that bypasses Geometrization.

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Submitted
Oct 21, 2019
Last revised
May 14, 2022 (v3)
Subjects
math.GT, math.DG
Journal
Compositio Math. 161 (2025) 447-502
DOI
10.1112/S0010437X24007565

Comments: 68 pages, 4 figures. Version 3: Rewritten introduction with focus on the applications to injectivity radius, diameter, and non-arithmeticity and non-commesurability of random 3-manifolds. Fundamentally reworked the proof concerning the model metrics

Geometrization3-manifolds

Its neighbourhood

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  1. Canonical geometrization of orientable -manifolds defined by vector-colourings of -polytopes

    Nikolai Erokhovets

    Topics Geometrization · Shared terms “geometrization conjecture”, “Perelman”

  2. A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture

    Yuan Liao, Ziyang Qin, Jui-Hui Chung et al.

    Topics Geometrization · Shared terms “geometrization conjecture”, “Perelman”

  3. Geometrization in Geometry

    Izabella Muraro de Freitas and Álvaro Krüger Ramos

    Topics Geometrization · Shared terms “Perelman”

  4. The entropy formula for the Ricci flow and its geometric applications

    Grisha PerelmanThe canon

    Topics Geometrization, 3-manifolds · Shared terms “geometrization conjecture”, “injectivity radius”

  5. Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings In Minkowski Space And The Cross Curvature Flow

    Paul Bryan, Mohammad N. Ivaki and Julian Scheuer

    Topics Geometrization, 3-manifolds · Shared terms “hyperbolic metrics”

  6. Geometrization of the local Langlands correspondence: an overview

    Laurent Fargues

    Topics Geometrization · Shared terms “geometrization conjecture”

  7. Large-scale geometry of the Universe

    Yassir Awwad and Tomislav Prokopec

    Topics Geometrization, 3-manifolds · Shared terms “geometrization conjecture”, “diameter”, “Perelman”

  8. An Introduction to Geometric Topology

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    Topics Geometrization, 3-manifolds · Shared terms “topological”, “Perelman”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.