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A rank-two horocusp truncation has finite volume

Statement

Let $Γ \subset C$ be a rank-two lattice acting on $H^{3}$ by horizontal translations, and let

C = {(x + i y, t) : t \geq 1} / Γ

be the quotient of the horoball $t \geq 1$ . If $A$ is the Euclidean area of a fundamental parallelogram for $Γ$ , then the end is at infinite hyperbolic distance and $Vol (C) = A / 2$ .

Statement references

Jessica S. Purcell, Hyperbolic Knot Theory, §4.3

Facts & Assumptions

Given: The upper half-space metric and a rank-two translation lattice $Γ$ .

[F1]

Ideal tetrahedron shape and cusp holonomy gives $d s^{2} = t^{- 2} (d x^{2} + d y^{2} + d t^{2})$ and its volume density $d x d y d t / t^{3}$ .

Proof

1.1givenF1algebra

Let $Q$ be a fundamental parallelogram of area $A$ . Integrating the density from [F1] over $Q \times [1, \infty)$ gives $Vol (C) = \int_{Q} \int_{1}^{\infty} t^{- 3} d t d x d y = A [- 1 / (2 t^{2})]_{1}^{\infty} = A / 2$ . The improper integral converges because $t^{- 3}$ is integrable on $[1, \infty)$ .

2.1givenF1algebra∎

Along any piecewise smooth path with height coordinate $t$ , its hyperbolic length is at least $\int ∣ d t ∣ / t$ , since the metric contains the term $d t^{2} / t^{2}$ . A path escaping the end has $t \to \infty$ , so this lower bound diverges as $\log t$ . Thus the end is at infinite distance and is complete toward its ideal end.

Sources

William P. Thurston, The Geometry and Topology of Three-Manifolds, Chapter 4

Depends on

Ideal tetrahedron shape and cusp holonomy

Used by

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Sources

Jessica S. Purcell, Hyperbolic Knot Theory (standard reference, not scraped)
William P. Thurston, The Geometry and Topology of Three-Manifolds (standard reference, not scraped)