Finite Width of the 14-Triangle Complex Group

Definitions and mathematical statement

Let $G$ be the group associated in the report with the finite 14-triangle complex, given by \[ G=\langle x,y\mid r_1,r_2,r_3\rangle, \] where the relators are exactly \[ r_1=yxyxyxy^{-3}x^{-3}, \qquad r_2=yx^{-1}y^{-1}x^{-3}y^2x^{-1}yxy, \qquad r_3=y^3x^{-1}yxyx^2y^2xyx. \] The original record denotes its lower central series by $\gamma_0(G)=G,\gamma_1(G),\ldots$, "as defined in the report." For every integer $i\geq2$, the proposed exact index pattern is \[ [\gamma_i(G):\gamma_{i+1}(G)]=8 \quad\text{if }i\equiv0\text{ or }1\pmod3, \] \[ [\gamma_i(G):\gamma_{i+1}(G)]=4 \quad\text{if }i\equiv2\pmod3. \] The source calls this equivalent to $G$ having finite width $3$.

The indexing convention is material: for an ordinary lower central series indexed from zero, one would set $\gamma_{i+1}(G)=[\gamma_i(G),G]$, with commutator subgroup generated by $a^{-1}b^{-1}ab$ for the indicated elements. The extraction does not reproduce the report's recursion, so this is an explicit possible interpretation, not a silent replacement by the common convention $\gamma_1(G)=G$. A shift of indices would change the residue classes in the conjecture.

The exact definition of width is also absent. Bounded width alone does not generally force this particular periodic pattern; any group-specific equivalence needs the report's context. If the relevant factors are elementary abelian $2$-groups and width is measured by their dimensions over ${\mathbb{F}}_2$, orders $8$ and $4$ correspond to dimensions $3$ and $2$ for the stated indices, but the factor structure and treatment of the initial terms are not supplied. No independent converse is asserted.

Short English statement

For the explicitly presented 14-triangle group, do successive series quotients have orders $8,8,4$ according to the stated index residues for every $i\geq2$? Preserve the report's zero-based series convention and verify its finite-width reformulation.

Sources

• Local UnsolvedMath registry, permanent ID $30001033$, source code catalogue entry, "Finite Width of the 14-Triangle Complex Group." No individual proposer is supplied.
• Geometric Group Theory, Hyperbolic Dynamics and Symplectic Geometry (2009), Oberwolfach Reports, DOI 10.4171/owr/2008/42, pp. 2337--2406: https://doi.org/10.4171/owr/2008/42.
• Supplied reference, "Cayley graph expanders and groups of finite width": https://arxiv.org/abs/0809.1560.
• Supplied reference, "Finite-width computations": https://arxiv.org/abs/1304.4480.
• Supplied literature triage, checked 2026-08-21: status "open." It reports that the primary paper still calls finite width three a conjecture and that later computations do not prove the general assertion. This is supplied dated curation, not independently verified; time-sensitive status and cited claims require independent verification.