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Parafree graphs of groups with cyclic edge groups

Andrei Jaikin-Zapirain, Ismael Morales

Abstract

We establish a combination theorem for parafree groups. These groups were introduced by Baumslag in the sixties. One of the current motivations for a better understanding of their structure is that they show up naturally in connection with Remeslennikov's conjecture on the profinite rigidity of free groups. In this article, we determine when the fundamental group of a finite graph of groups with cyclic edge groups is parafree.

Parafree graphs of groups with cyclic edge groups

Abstract

We establish a combination theorem for parafree groups. These groups were introduced by Baumslag in the sixties. One of the current motivations for a better understanding of their structure is that they show up naturally in connection with Remeslennikov's conjecture on the profinite rigidity of free groups. In this article, we determine when the fundamental group of a finite graph of groups with cyclic edge groups is parafree.

Paper Structure

This paper contains 15 sections, 20 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General notation for rings, groups and pro-$p$ groups
  4. Parafree groups
  5. Group algebra and the augmentation ideal
  6. The fundamental groups of graphs of groups
  7. The induced map between the augmentation ideals
  8. $\mathop{\mathrm{\mathcal{D}}}\nolimits$-torsion-free modules
  9. The universal division rings of fractions of the group algebra of a subgroup of a free pro-$p$ group
  10. Embeddings of abstract groups into free pro-$p$ groups
  11. Examples of $\mathop{\mathrm{\mathcal{D}}}\nolimits_{\mathbb{F}_p G}$-torsion-free modules
  12. Proofs of the main results
  13. Amalgamated products
  14. HNN extensions
  15. The fundamental group of graph of groups

Key Result

Theorem 1.1

Let $U$ and $V$ be finitely generated groups, $1\neq u\in U$ and $1\neq v\in V$. Consider the amalgamated free product $W=U\underset{u=v}{*}V$. Then $W$ is parafree if and only if the following three conditions hold.

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • ...and 44 more