Positive laws on generators in powerful pro-p groups

TL;DR

This paper proves that certain finitely generated powerful pro-p groups satisfying specific laws are nilpotent, with bounds on their nilpotency class, especially when verbal subgroups satisfy positive laws.

Contribution

It establishes a new link between positive laws, verbal subgroups, and nilpotency in powerful pro-p groups, providing explicit bounds on nilpotency class.

Findings

Powerful pro-p groups satisfying a law are nilpotent.

Verbal subgroups satisfying positive laws are nilpotent.

Bounds on nilpotency class depend on prime p, generators, and laws.

Abstract

If G is a finitely generated powerful pro-p group satisfying a certain law v=1, and if G can be generated by a normal subset T of finite width which satisfies a positive law, we prove that G is nilpotent. Furthermore, the nilpotency class of G can be bounded in terms of the prime p, the number of generators of G, the law v=1, the width of T, and the degree of the positive law. The main interest of this result is the application to verbal subgroups: if G is a p-adic analytic pro-p group in which all values of a word w satisfy positive law, and if the verbal subgroup w(G) is powerful, then w(G) is nilpotent.