Kourovka Notebook Problem 21.69

Definitions and mathematical statement

A finite one-relator presentation has the form \[ G=\langle x_1,\ldots,x_n\mid R\rangle =F_n/\langle\!\langle R\rangle\!\rangle, \] where $n$ is finite, $F_n$ is the free group on the listed generators, $R$ is a finite word in them and their inverses, and the double angle brackets denote the normal subgroup generated by $R$. The input consists of this finite list and word. No restriction is imposed on the number of occurrences of a generator in the relator, and proper-power relators are allowed.

Give the Cayley graph of $G$ for the indicated symmetric generating set edges of length one and its induced path metric. The group is word-hyperbolic if some real $\delta\ge0$ has the following property: every side of every geodesic triangle lies within distance $\delta$ of the union of the other two sides. This property is independent of the chosen finite generating set.

Is there an algorithm that, for every finite one-relator presentation, halts and decides whether its presented group is word-hyperbolic? The algorithm must handle both positive and negative instances; a procedure that eventually verifies hyperbolicity when it holds but may run forever otherwise does not suffice.

The question asks for one uniform algorithm over all such presentations, not a decision procedure tailored to a single relator. No value of $\delta$ is supplied as part of the input, and no running-time bound is required. Finite groups and other elementary hyperbolic groups are included in the positive class.

Short English statement

Can one algorithm decide word-hyperbolicity from every finite one-relator presentation?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2578 (KOU-21.69), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.69. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.