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McMullen's injectivity-radius conjecture for hyperbolic 3-manifolds

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Let GG be a finitely generated group with gg generators. For a hyperbolic 33-manifold NN, write inj⁡N(x)\operatorname{inj}_N(x) for the injectivity radius at xx, and let the convex core be the minimal convex submanifold whose inclusion is a homotopy equivalence. McMullen's injectivity-radius conjecture. There exists a constant C=C(g)C=C(g) such that, whenever π1(N)\pi_1(N) is isomorphic to GG and xx lies in the convex core of NN,

inj⁡N(x)≤C.\operatorname{inj}_N(x)\leq C.

The conjecture predicts a uniform upper bound, depending only on the number of generators, for injectivity radius throughout the convex core. The source attributes it to McMullen and gives no resolution status.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. McMullen's injectivity-radius conjecture for hyperbolic 3-manifolds

    Let MM be a closed hyperbolic 33-manifold, let kk be a positive integer, and let inj⁡(M,x)\operatorname{inj}(M,x) denote the injectivity radius at x∈Mx\in M. McMullen's conjecture. For every kk there is some RR such that

    inj⁡(M,x)≤R\operatorname{inj}(M,x)\le R

    for all xx in every closed hyperbolic 33-manifold MM with rank⁡(π1(M))=k\operatorname{rank}(\pi_1(M))=k. This conjecture bounds the radius of the largest embedded ball in terms of the rank of the fundamental group; the source notes a generalization to infinite-volume manifolds and a reduction of that case to the closed case. Its resolution is not specified here.

    source: Juan Souto, “Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds”, arXiv:0904.0237 (2009).

Sources & referencesView supporting material

Primary source

James W. Anderson, “A brief survey of the deformation theory of Kleinian groups”, arXiv:math/9810186 (1998).

3-manifoldsfundamental-groupshyperbolic-3-manifoldshyperbolic-geometrykleinian-groups

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