Matrix stability of crystallographic groups
Some years ago, I proved with Shulman and Sørensen that precisely 12 of the 17 wallpaper groups are matricially stable in the operator norm. We did so as part of a general investigation of when group C∗C∗-algebras have the semiprojectivity and weak matricial semiprojectivity properties — notions which are standard tools in the classification theory for C∗C∗-algebras.
Our results were largely negative, and recently Dadarlat has provided a framework for understanding the obstructions to matricial stability for discrete groups. With this perspective, our results may be seen as showing that, at least in this case, stability ensues when the obstructions allow it.
I intend to go through the proof of this positive result in a form aimed at non-C∗C∗-algebraists. It must be admitted that the proof is very C∗C∗-algebraic in nature, but it goes through a natural dimension reduction technique (invented by Friis and Rørdam in the mid-90’s) which I can definitely explain and expect could be useful in other settings as well.