# Infinitely generated powerful pro-$p$ groups

A pro-$p$ group of finite subgroup rank has an open subgroup $P$ that is uniformly powerful, meaning that $[P,P]$ is contained in the group generated by $2p$-th powers in $P$, and raising elements to the power $p$ induces an isomorphism between successive factors of the lower central $p$-series of $P$. This is the key ingredient in the theory of $p$-adic Lie groups as developed by Lubotzky and Mann.

For a general pro-$p$ group $G$, there is no reason for $G$ to have any large uniformly powerful subgroups. But the definition of uniformly powerful still makes sense for groups that are not topologically finitely generated. How much is known about uniformly powerful groups without the assumption of finite generation? Alternatively, what about pro-$p$ groups in which every finitely generated subgroup has finite subgroup rank?

-