Inertia of retracts in Demushkin groups

TL;DR

This paper proves that all retracts of Demushkin groups are inert, confirming a conjecture related to the structure and properties of these pro-p groups, based on inequalities involving rank and relation gradients.

Contribution

It establishes that every retract of a Demushkin group is inert, advancing the understanding of subgroup structure in pro-p groups and confirming a key conjecture.

Findings

All retracts of Demushkin groups are inert.

Supports the Dicks-Ventura Inertia Conjecture for Demushkin groups.

Builds on inequalities of rank and relation gradients in pro-p modules.

Abstract

Exploring inequalities regarding the rank and relation gradients of pro-p modules and building upon recent results of Y. Antol\'in, A. Jaikin-Zapiran and M. Shusterman, we prove that every retract of a Demushkin group is inert in the sense of the Dicks-Ventura Inertia Conjecture.