Ascending chains of finitely generated subgroups

TL;DR

This paper proves that families of finitely generated subgroups in pro-$p$ groups have maximal elements and uses Noetherian induction to reveal new properties of these subgroups, including the behavior of their commensurators and ascending chains.

Contribution

It introduces the use of Noetherian induction in the context of finitely generated subgroups of pro-$p$ groups and establishes new results on the structure of ascending chains and commensurators.

Findings

Families of n-generated subgroups have maximal elements.

The commensurator of a finitely generated subgroup is the largest containing subgroup.

Ascending chains of n-generated subgroups in limit groups terminate.

Abstract

We show that a nonempty family of $n$-generated subgroups of a pro-$p$ group has a maximal element. This suggests that 'Noetherian Induction' can be used to discover new features of finitely generated subgroups of pro-$p$ groups. To demonstrate this, we show that in various pro-$p$ groups $\Gamma$ (e.g. free pro-$p$ groups, nonsolvable Demushkin groups) the commensurator of a finitely generated subgroup $H \neq 1$ is the greatest subgroup of $\Gamma$ containing $H$ as an open subgroup. We also show that an ascending sequence of $n$-generated subgroups of a limit group must terminate (this extends the analogous result for free groups proved by Takahasi, Higman, and Kapovich-Myasnikov).