Kirby Problem 3.2

Definitions and mathematical statement

Let $\mathcal V$ be the set of volumes of all connected complete finite-volume hyperbolic three-manifolds, with sectional curvature normalized to $-1$. Volumes are positive real numbers, and a finite covering of degree $d$ multiplies volume by $d$. In particular, manifolds sharing a finite-sheeted cover have rationally related volumes. The problem asks for rational independence beyond this necessary relation.

The phrase "not all rationally related" has two versions explicitly distinguished in the source. The first asks for manifolds $M,N$ with \[ \frac{\operatorname{vol}(M)}{\operatorname{vol}(N)}\notin\mathbb Q. \] The stronger version asks whether the vector space $\operatorname{span}_{\mathbb Q}\mathcal V$ is infinite-dimensional. Equivalently, for every positive integer $r$ there should be manifolds $M_1,\ldots,M_r$ such that \[ \sum_{j=1}^r q_j\operatorname{vol}(M_j)=0, \quad q_j\in\mathbb Q \quad\Longrightarrow\quad q_1=\cdots=q_r=0. \] These assertions are recorded separately; an irrational ratio only proves that the dimension is at least two. Likewise, proving that one volume is irrational does not alone prove that its ratio with a second volume is irrational.

Short English statement

Prove that some hyperbolic three-manifold volumes have irrational ratios; more strongly, show that their rational span has arbitrarily large dimension.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2800 (KP-3.2), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 3.2. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.