Kirby Problem 3.31

Definitions and mathematical statement

A nontrivial group $G$ is left-orderable if it admits a strict total order $<$ satisfying $a<b\Rightarrow ga<gb$ for every $a,b,g\in G$. Equivalently there is a positive cone $P\subset G$ with \[ PP\subseteq P,\qquad G=P\sqcup\{1\}\sqcup P^{-1}. \] The displayed union is disjoint. Following the source, exclude the trivial group from the property; its alternative conventional treatment changes only that special input case.

Given a finite triangulation of a closed three-manifold $M$, can an algorithm decide whether $\pi_1(M)$ is left-orderable? It must halt on both positive and negative instances, with the triangulation providing a finite presentation for the fundamental group.

Also determine the complexity of certificates of left-orderability or non-left-orderability. A certificate means finite data whose validity can be checked from the input; the desired size and checking time should be stated as functions of its length. Describing an arbitrary infinite positive cone without a finite effective rule is not a finite certificate. One may also study the complexity of recognizing positive words for a specified order, but this is different from deciding whether any left order exists.

Short English statement

Decide left-orderability for closed three-manifold groups and understand how efficiently either answer can be certified.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2829 (KP-3.31), 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 3.31. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.