Induction of path-equivalence by G_n^3 relations in the realisable image

Determine whether the equivalence relation on the image φ(Path(𝔛₁)/⟨∼⟩), obtained by mapping homotopy classes of paths in the flip graph 𝔛₁ of rhombile tilings of a 2n-vertex zonogon to the group G_n^3 by assigning to each flip on a cube generated by e_i, e_j, e_k the generator a_{ijk}, is induced by the defining relations of G_n^3 (involutions, far-commutativity, and the octagon relation).

Background

The paper constructs a flip graph 𝔛₁ whose vertices are rhombile tilings of a zonogon with 2n vertices and whose edges represent flips (stackings of 3D cubes). A 2-dimensional cell complex 𝔛₂ is obtained by attaching 2-cells corresponding to far-commutativity and the octagon relation, and 𝔛₂ is shown to be simply connected.

A map φ from homotopy classes of paths in 𝔛₁ (relative endpoints, modulo the 2-cells in 𝔛₂) to G_n^3 is defined by sending each flip on a cube generated by indices {i,j,k} to the generator a_{ijk} of G_n^3 and then reading the product along the path. This map is well-defined because the 2-cells mirror the defining relations of G_n^3.

The authors introduce the "realisable counterpart" φ(Path(𝔛₁)/⟨∼⟩) ⊂ G_n^3. While inverses exist for elements arising from paths and composition corresponds to concatenation of composable paths, the group structure on the image is unclear. Specifically, it is unknown whether the equivalence relation on the image induced from path homotopies is the same as the one induced by the relations in G_n^3, which would clarify the subgroup structure.