Character degrees and nilpotence class of finite $p$-groups: An approach via pro-$p$ groups

Let $\mathcal{S}$ be a finite set of powers of $p$ containing 1. It is known that for some choices of $\mathcal{S}$, if $P$ is a finite $p$-group whose set of character degrees is $\mathcal{S}$, then the nilpotence class of $P$ is bounded by some integer that depends on $\mathcal{S}$, while for some other choices of $\mathcal{S}$ such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set $\mathcal{S}$ is class bounding if and only if $p\notin\mathcal{S}$. In this article we provide a new approach to this problem. Our main result shows the relevance of certain $p$-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets $\mathcal{S}$ such that $p\notin\mathcal{S}$.