MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.