I apologize in advance for a somewhat wishy-washy question. I just read the paper "The Braid Group and Other Groups" by F. A. Garside in which he solves the conjugacy problem for the braid group and also obtains a standard form for braids that yields a solution to the word problem and computes the center. The paper is straightforward to follow line by line, however the last section "Other groups" makes me feel like I am completely missing the geometric insight that Garside had while writing the paper - in particular he says that all of the results given above hold for a certain class of more general groups to give the solution to the word/conjugacy problems. He also notes that the diagram that he has drawn for the element $\Delta \in B_4$ is the 2-skeleton of the truncated octahedron and the are similar higher-dimensional polytopes for the other elements $\Delta \in B_n$.
For example, corresponding to the truncated cuboctahedron, Garside tells us that for the group generated by $a_1, a_2, a_3$ subject to the relations $a_1a_2a_1a_2 = a_2 a_1a_2a_1, a_2a_3a_2= a_3a_2a_3, a_1a_3=a_3a_1$, if we consider the element $\Delta = (a_1a_2a_3)^3$, then the results proved in Garside's paper, all suitably modified, still hold. How is this group related to that polytope?
Can someone guide me with the geometric rehydration process for this paper? For concreteness, this might involve relating some of the results in the paper to constructions with a polytope.