[Paper Review] Injectivity Radius and Fundamental Group of Hyperbolic 3-Manifolds

Matthew E. White|ArXiv.org|Apr 19, 2001

Geometric and Algebraic Topology3 references3 citations

TL;DR

This paper establishes that for any closed hyperbolic 3-manifold with fundamental group of rank $ n > 1 $, the injectivity radius is universally bounded above by a constant $ R_n $, independent of the specific manifold. The proof uses minimal-length trivalent graphs carrying the fundamental group and shows that if the injectivity radius were too large, relations in the group would force a shorter such graph, contradicting minimality and implying the group is free—impossible for a closed hyperbolic 3-manifold due to irreducibility.

ABSTRACT

We show that there is an upper bound on the injectivity radius of a hyperbolic 3-manifold in terms of the the number of generators of its fundamental group.

Motivation & Objective

• To establish an upper bound on the injectivity radius of closed hyperbolic 3-manifolds in terms of the rank of their fundamental group.
• To show that such a bound exists universally for each rank $ n > 1 $, independent of the specific manifold.
• To use geometric and topological arguments involving minimal-length carrier graphs to derive a contradiction if the injectivity radius exceeds a threshold.
• To demonstrate that the fundamental group cannot be free, thus contradicting the existence of a minimal-length graph under large injectivity radius.
• To extend the result to manifolds with bounded negative curvature, generalizing the injectivity radius bound.

Proposed method

• Construct a minimal-length trivalent graph $ \Gamma $, called an $ n $-graph, with $ 3(n-1) $ edges, that carries the fundamental group of the manifold.
• Use the universal cover $ \mathbb{H}^3 \to M $ to lift the graph and analyze edge lengths and proximity under the assumption of large injectivity radius.
• Apply a geometric argument showing that if edges are close in $ \mathbb{H}^3 $, they can be replaced via a shortcut arc to reduce total length while preserving group-carrying property.
• Use the fact that any loop in $ \Gamma $ must map to a nontrivial loop in $ M $, so its length exceeds $ 2 \cdot \text{inj}(M) $, to constrain edge lengths.
• Apply residual finiteness of hyperbolic 3-manifolds to construct finite-sheeted covers with arbitrarily large fundamental group rank.
• Use contradiction: if $ \text{inj}(M) \geq R_n $, then a shorter carrier graph exists, implying $ \pi_1(M) $ is free, which contradicts irreducibility.

Experimental results

Research questions

• RQ1 Can the injectivity radius of a closed hyperbolic 3-manifold be bounded above in terms of the rank of its fundamental group?
• RQ2 Is there a universal constant $ R_n $ such that any closed hyperbolic 3-manifold with $ \text{Rank}(\pi_1(M)) = n $ must satisfy $ \text{inj}(M) < R_n $?
• RQ3 What geometric or topological obstruction prevents the existence of a minimal-length carrier graph when the injectivity radius is too large?
• RQ4 Can the fundamental group of a closed hyperbolic 3-manifold be free, and how does this relate to injectivity radius bounds?
• RQ5 Does residual finiteness allow the construction of finite-sheeted covers with arbitrarily large fundamental group rank?

Key findings

• For each integer $ n > 1 $, there exists a universal constant $ R_n > 0 $ such that any closed hyperbolic 3-manifold $ M $ with $ \text{Rank}(\pi_1(M)) = n $ must satisfy $ \text{inj}(M) < R_n $.
• The proof relies on the existence of a minimal-length trivalent graph $ \Gamma $ carrying $ \pi_1(M) $, with $ 3(n-1) $ edges, and the fact that such a graph cannot exist if $ \text{inj}(M) $ is too large.
• If $ \text{inj}(M) $ were large enough, edges of the graph would be forced into close proximity in $ \mathbb{H}^3 $, allowing a shortcut that reduces total length while preserving the group-carrying property.
• This leads to a contradiction because it would imply $ \pi_1(M) $ is free, but closed hyperbolic 3-manifolds are irreducible and cannot have free fundamental groups.
• Residual finiteness of hyperbolic 3-manifolds implies the existence of finite-sheeted covers $ \tilde{M} \to M $ with arbitrarily large $ \text{Rank}(\pi_1(\tilde{M})) $.
• The result extends to manifolds with bounded negative curvature, not just hyperbolic ones.