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Revisiting fully residually free Demushkin groups

Henrique Souza, Pavel Zalesskii

Abstract

We establish new examples and non-examples of pro-p limit groups among the class of Demushkin groups, that is, pro-p Poincaré duality groups of dimension 2.

Revisiting fully residually free Demushkin groups

Abstract

We establish new examples and non-examples of pro-p limit groups among the class of Demushkin groups, that is, pro-p Poincaré duality groups of dimension 2.
Paper Structure (3 sections, 13 theorems, 25 equations, 1 figure)

This paper contains 3 sections, 13 theorems, 25 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries on pro-p groups acting on pro-p trees
  3. Demushkin groups

Key Result

Theorem 1.3

Let $G$ be a pro-$p$ Demushkin group with minimal number of generators $n$, torsion-invariant $q \in \{0\,,p\,,p^2\,,\cdots\}$ and orientation character $\chi\colon G \to \mathbb{Z}_p^\times$. Moreover, in both cases above, $G$ is (fully) residually free pro-$p$.

Figures (1)

  • Figure 1: What happens in $\Gamma$ when a terminal vertex has two incident edges in $\Delta$. Filled in vertices are in the image of $\sigma$, hollowed out vertices may be not. Left picture has the labels induced from the map $\sigma$, right picture shows the labels as $G$-cosets of $A$, $C$ and $G_{n-1}$.

Theorems & Definitions (26)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1: chatzidakisPropGroupsActing2022
  • Proposition 2.2: snopceSubgroupPropertiesPro$p$2014
  • Proposition 3.1: neukirchCohomologyNumberFields2008
  • Definition 3.2
  • Theorem 3.3: labuteClassificationDemushkinGroups1967
  • Theorem 3.4
  • proof
  • Remark 3.5
  • ...and 16 more