Open Problems and Questions about Geodesics — Conjecture 11 (Hingston)

Definitions and mathematical statement

The supplied record contains only a fragment of Conjecture 11. It uses a parameter $g$, a coefficient label $G$, a quantity $\alpha_{g,G}$, and quantities $a,b,c$, without defining them or specifying their domains or quantifiers. The displayed fragment is \[ \begin{aligned} \alpha_{g,G}&=a &&\text{if }G=\mathbb{Q},\\ \alpha_{g,G}&=b &&\text{if }G=\mathbb{Z}_{3},\\ \alpha_{g,G}&=c &&\text{if }G=\mathbb{Z}_{7},\\ \alpha_{g,G}&=a &&\text{if }G=\mathbb{Z}_{p},\quad p\neq 3,7. \end{aligned} \] Here $\mathbb{Q}$ denotes the rational numbers. The source does not explicitly define $\mathbb{Z}_{p}$ or specify the permitted values of $p$; in particular, a restriction to prime $p$ is not supplied. No geometric setting, definition of $\alpha_{g,G}$, restrictions on $a,b,c$, or assertion of existence or universality is provided. Thus these relations preserve the supplied fragment but do not constitute a fully specified standalone conjecture.

Short English statement

The fragment assigns the same value $a$ to $\alpha_{g,G}$ for rational coefficients and for the listed $\mathbb{Z}_{p}$ cases with $p\neq 3,7$, while assigning values $b$ and $c$ for $\mathbb{Z}_{3}$ and $\mathbb{Z}_{7}$. Its definitions, hypotheses, and quantifiers are missing.

Sources

• Local UnsolvedMath registry, permanent ID $800931$, exact source problem code catalogue entry; supplied title: Open Problems and Questions about Geodesics --- Conjecture 11 (Hingston). The registry labels the problem open; this is a supplied status, not an independently verified mathematical assessment.
• Burns and Matveev, Open Problems and Questions about Geodesics, listed in the supplied record as a 2019 source, Conjecture 11: https://arxiv.org/abs/1308.5417. The registry lists Burns and Matveev as proposers and separately records the attribution/context as Hingston; the precise attribution is not explained.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance notes; the linked source and extraction have not been independently checked for this draft.