For any finite group $G$, we can define a 'dimension' of the group as the smallest $n$ such that there is some choice of generators $S$ of $G$ where $G$ equipped with word metric with respect to $S$ is isometrically embeddable into euclidean space $\mathbb{R}^n$. Equivalently, this means the cayley graph on $G$ (equipped with shortest-path-length metric) with edges labelled by elements in $S$ and edge-lengths $1$ is isometrically embeddable into $\mathbb{R}^n$. This is a finite number, since taking $S = G$ you get the discrete metric, which is embeddable in $\mathbb{R}^{|G|-1}$ as a simplex.
- Is there a name for this 'dimension'? I'm not having any luck with google searches.
- I'd be interested to know bounds on this dimension (assuming it is studied as is nontrivial) for the finite simple groups