Post History

Proving that this relation implies another relation on the Coxeter group [4,3,3,4].

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)^3 = \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.