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Profinite detection of free products and free factors

Andrei Jaikin-Zapirain, Henrique Souza, Pavel Zalesski

Abstract

Let $G$ be the fundamental group of a graph of finitely generated virtually free groups with virtually cyclic edge groups. We shaw that $G$ is cohomologically good if $G$ is residually finite. If $G$ is LERF, we prove that G splits non-trivially as a free product if and only if its profinite completion $\widehat{G}$ splits non-trivially as a free profinite product. Moreover, we are able to detect one-ended free factors of $G$ from $\widehat{G}$. As an application, we deduce that any profinitely rigid word in a finitely generated free group is universally profinitely rigid.

Profinite detection of free products and free factors

Abstract

Let be the fundamental group of a graph of finitely generated virtually free groups with virtually cyclic edge groups. We shaw that is cohomologically good if is residually finite. If is LERF, we prove that G splits non-trivially as a free product if and only if its profinite completion splits non-trivially as a free profinite product. Moreover, we are able to detect one-ended free factors of from . As an application, we deduce that any profinitely rigid word in a finitely generated free group is universally profinitely rigid.
Paper Structure (17 sections, 34 theorems, 68 equations)

This paper contains 17 sections, 34 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. The fundamental group of a graph of groups
  5. Derivations
  6. The profinite fundamental group of a graph of profinite groups
  7. The class C
  8. Residually finiteness and LERFness in the class C
  9. On a result of Wilton
  10. Fixed points of profinite actions
  11. Flatness of the completed algebra of the profinite completion
  12. Small cancellation groups
  13. The profinite completion of finitely generated residually finite groups in C
  14. Relative splitting theorem for virtually free groups
  15. Detecting free products and free factors
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ be a finitely generated LERF group that decomposes as the fundamental group of a graph of virtually free groups with virtually cyclic edge groups. If $\widehat{G}$ decomposes as a non-trivial free profinite product, then $G$ decomposes as a non-trivial free product.

Theorems & Definitions (71)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3: RibesZalesskii2010
  • Lemma 2.4
  • ...and 61 more