Kirby Problem 3.37

Definitions and mathematical statement

A three-manifold group is a group isomorphic to $\pi_1(M)$ for a connected three-manifold $M$. Consider those which are finitely generated. A group $G$ is linear over a characteristic-zero field if there are a field $k$ with $\operatorname{char}(k)=0$, an integer $d\geq1$, and an injective homomorphism \[ \rho:G\hookrightarrow\operatorname{GL}_d(k). \] Faithfulness means that the only element represented by the identity matrix is the identity of $G$; an arbitrary nontrivial matrix representation is not sufficient.

(a) Is every finitely generated three-manifold group linear over some characteristic-zero field? Both the field and dimension may depend on $G$.

(b) If so, what bounds can be given for the minimum possible dimension $d_0(G)$? One can seek a universal bound for all such groups, or an effective bound in terms of specified finite manifold data, and these versions must be distinguished. A bound obtained for a single geometric piece is not by itself a bound for the full manifold group after gluing its pieces.

The characteristic-zero qualification is part of the problem. Representations over fields of positive characteristic alone do not establish (a), and the injectivity requirement concerns all elements, not just a chosen generating set.

Short English statement

Are all finitely generated three-manifold groups faithful matrix groups in characteristic zero, and how large a matrix dimension is necessary?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2835 (KP-3.37), 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 3.37. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.