[Paper Review] Presentation length and Simon's conjecture

Ian Agol, Yi Liu|arXiv (Cornell University)|Jun 28, 2010

Geometric and Algebraic Topology20 references4 citations

TL;DR

This paper proves that any finitely generated group with first Betti number one maps onto at most finitely many knot groups, confirming a conjecture by J. Simon. The authors establish a linear diameter bound for closed hyperbolic 3-manifolds in terms of the presentation length of their fundamental group, using factorization through extended Dehn fillings and coherence of fundamental groups in geometric pieces.

ABSTRACT

In this paper, we show that any knot group maps onto at most finitely many knot groups. This gives an affirmative answer to a conjecture of J. Simon. We also bound the diameter of a closed hyperbolic 3-manifold linearly in terms of the presentation length of its fundamental group, improving a result of White.

Motivation & Objective

• To resolve J. Simon's conjecture that any knot group maps onto at most finitely many other knot groups.
• To bound the diameter of closed hyperbolic 3-manifolds linearly in terms of the presentation length of their fundamental group.
• To develop a factorization technique through extended Dehn fillings to overcome obstructions in homotopy lifting for short geodesics.
• To extend results from hyperbolic knot complements to general knot complements using JSJ decomposition and case-by-case analysis.
• To provide a framework for understanding epimorphisms from a fixed group to 3-manifold groups, particularly knot groups with b1=1.

Proposed method

• Use presentation length, defined by Cooper, as a coarse substitute for simplicial volume to bound geometric invariants.
• Apply extended Dehn fillings (Dehn extensions) to factor epimorphisms through the fundamental group of a drilled manifold, ensuring coherence.
• Utilize the Scott core theorem and coherence of fundamental groups in geometric pieces to derive contradictions when epimorphisms factor through covers of extended drilling spaces.
• Bound the diameter of the thick part using [7, Theorem 0.1], which relates volume to presentation length.
• Bound the diameter of thin tubes by controlling the tube radius via the inequality π sinh²(r) ≤ A(ℓ(G)), where A(n) = 27ⁿ(9n² + 4n)π.
• Use homology obstruction: if φ factors through π₁(Nᵉ), then H₃(Nᵉ; ℚ) = 0, but the composition induces an isomorphism on H₃(–; ℚ) ≅ ℚ, leading to contradiction unless the tube radius is bounded.

Experimental results

Research questions

• RQ1 Can a finitely generated group G with b₁(G) = 1 map onto infinitely many knot groups?
• RQ2 Is there a linear upper bound on the diameter of a closed hyperbolic 3-manifold in terms of the presentation length of its fundamental group?
• RQ3 Can epimorphisms from a group G to knot groups be factored through extended Dehn fillings of geometric pieces, especially when the image group has a short geodesic?
• RQ4 Does the JSJ decomposition of a knot complement allow for a finite classification of possible image groups under epimorphisms from a fixed G with b₁(G) = 1?
• RQ5 Can the results be extended to knot complements in rational homology spheres, or are they restricted to S³?

Key findings

• Any knot group maps onto at most finitely many other knot groups, confirming Simon’s conjecture.
• The diameter of a closed hyperbolic 3-manifold is bounded linearly by the presentation length ℓ(G) of its fundamental group: diam(M) < C·ℓ(G) for some universal constant C > 0.
• The tube radius r of a short geodesic in a hyperbolic knot complement satisfies r < C₂·ℓ(G), with C₂ depending only on the presentation length.
• The bound on the tube radius arises from the homology obstruction: if the epimorphism φ: G → π₁(M) factors through an extended drilling space Nᵉ, then H₃(Nᵉ; ℚ) = 0, contradicting the fact that the composition induces an isomorphism on H₃(–; ℚ) ≅ ℚ.
• The simplicial volume of the image manifold is bounded in terms of the presentation length ℓ(G), providing a coarse substitute for volume bounds when simplicial volume vanishes.
• The result holds even for knot complements in rational homology spheres, provided the JSJ decomposition and companion structure are controlled.