Linear volume bounds for drilling and filling

We prove uniform linear bounds on the volume variation under drilling and filling operations on finite volume hyperbolic 3-manifolds.

💡 Research Summary

The paper investigates quantitative volume changes that occur when performing two fundamental operations on finite‑volume hyperbolic 3‑manifolds: drilling out a simple closed geodesic and Dehn filling a cusp. While it is well‑known that drilling increases volume and filling decreases it, previous results (e.g., Bridgeman, Agol, ASTD) gave bounds of the form
 vol(M − γ) ≤ f(R)·vol(M) + g(R)·ℓ,
where f(R) > 1 is a multiplicative constant depending on the tube radius R around the geodesic γ, and g(R) is another function of R. The novelty of this work is to eliminate the multiplicative factor entirely and replace it with an additive, linear term that depends only on the local geometry of the geodesic or cusp.

Main Results

• Theorem A (Drilling): Let M be a closed hyperbolic 3‑manifold and γ ⊂ M a simple closed geodesic of length ℓ. Assume γ admits an embedded tubular neighbourhood of radius R ∈ (0,1). Then there exists a universal constant c > 0 such that the unique hyperbolic metric on M − γ satisfies
   vol(M − γ) ≤ vol(M) + c·ℓ·R.
  Thus the volume increase is at most linear in the product of the geodesic length and its tube radius.

• Corollary B: If γ is the shortest non‑trivial closed geodesic in M, then a standard Margulis‑type argument guarantees a tube radius R ≥ R₀ > 0 depending only on the dimension. Consequently, the volume increase is bounded by c·ℓ, i.e. purely linear in the length of the shortest geodesic.

• Theorem C (Filling): Let M be a finite‑volume hyperbolic 3‑manifold with a single cusp C bounded by a horospherical torus ∂C of area A. Choose a simple closed curve μ on ∂C (with respect to the flat metric) of length ℓ > √(cπ). The Dehn filling of M along μ, denoted M_μ, carries a hyperbolic metric satisfying
   vol(M_μ) ≤ vol(M) − A/(2π²ℓ²)·(1 − cπ²/ℓ²).
  Hence the volume loss is essentially proportional to 1/ℓ², with an explicit additive correction term.

Proof Strategy
The authors’ central tool is the σ‑invariant, defined for a closed 3‑manifold M as
 σ(M) = sup_{