I have a group with five generators $\sigma_i$, and the following relations:

\begin{split} \sigma_i^2 = \varepsilon \\ |i-j| \neq 1 \implies (\sigma_i\sigma_j)^2 = \varepsilon \\ (\sigma_0\sigma_1)^4 = \varepsilon \\ (\sigma_1\sigma_2)^3 = \varepsilon \\ (\sigma_2\sigma_3)^3 = \varepsilon \\ (\sigma_3\sigma_4)^4 = \varepsilon \\ (\sigma_0\sigma_1\sigma_2\sigma_3\sigma_4\sigma_3\sigma_2\sigma_1)^n = \varepsilon \\ \end{split}

Note that without the last relation this is the Coxeter group $[4,3,3,4]$. So we can think of this as the Coxeter group $[4,3,3,4]$ with an extra relation.

The task is to prove:

$$ (\sigma_0\sigma_4\sigma_1\sigma_3\sigma_2\sigma_1\sigma_3)^{2n} = \varepsilon $$

Why do I believe this could be true?

I've use the GAP system to confirm that this is true for $n < 16$. Some of these groups while always finite begin to get very large, so I think it's very likely this holds for all $n$.

What have I done so far?

Besides confirming it holds for cases, I've tried a couple of approaches.

• The first was to simply hope that $\sigma_0\sigma_1\sigma_2\sigma_3\sigma_4\sigma_3\sigma_2\sigma_1 = (\sigma_0\sigma_4\sigma_1\sigma_3\sigma_2\sigma_1\sigma_3)^2$ as if I could prove that, it would solve the problem. It turns out that is not the case.

• Next I hoped that $\sigma_0\sigma_1\sigma_2\sigma_3\sigma_4\sigma_3\sigma_2\sigma_1$ and $(\sigma_0\sigma_4\sigma_1\sigma_3\sigma_2\sigma_1\sigma_3)^2$ were conjugates in $[4,3,3,4]$, since conjugates have the same order in a group. GAP can solve for this and it found they were in fact not conjugates.

Where does this problem come from?

This problem sort of looks like random nonsense. Just something made from bashing together random generators until something stuck. However it is part of a larger problem which I will try to briefly justify here.

The Coxeter group $[4,3,3,4]$ corresponds to the symmetry of the 5-dimensional hypercubic honeycomb, and the additional relation that we give gives a subsymmetry of the 4-torus in 8-dimensional Euclidean space. The 4-torus being a quotient of 4-space.

The relation I am aiming to prove shows that this symmetry has a nice relationship to similar symmetries of the 2-torus in 4-space.

This relationship is useful to me for proving that certain types of polyhedral embeddings exist in 8-space.