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Absolute profinite rigidity, direct products, and finite presentability

M. R. Bridson, A. W. Reid, R. Spitler

TL;DR

This work investigates the extent to which finitely presented, residually finite groups are determined by their finite quotients, establishing the first infinite family of finitely presented groups with profinite rigidity among finitely presented groups but not among all finitely generated groups. The authors build Γ from Seifert fibred space groups with base orbifolds $S^2(p,q,r)$ drawn from explicit lists, and show that Γ×Γ is profinitely rigid in the class of finitely presented residually finite groups, while there exist infinitely many non-isomorphic finitely generated groups Λ with $\widehat{Λ}\cong\widehat{Γ}\times\widehat{Γ}$ whose embeddings into Γ×Γ yield Grothendieck pairs. The strategy blends Galois rigidity for triangle groups, central extensions, and a pro-finite Grothendieck framework, augmented by new constructions in products of central extensions of hyperbolic groups to produce infinitely many nonisomorphic profinitely-trivial quotients. The results illuminate how finite presentability constrains profinite phenomena and provide a wide supply of Grothendieck pairs in products, contributing to a deeper understanding of profinite genus and rigidity in 3-manifold groups with arithmetic structure.

Abstract

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form $Γ\times Γ$ where $Γ$ is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if $P$ is a finitely generated, residually finite group with $\widehat{P}\cong\widehat{Γ\timesΓ}$ then there is an embedding $P\hookrightarrowΓ\timesΓ$ that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for $P$.

Absolute profinite rigidity, direct products, and finite presentability

TL;DR

This work investigates the extent to which finitely presented, residually finite groups are determined by their finite quotients, establishing the first infinite family of finitely presented groups with profinite rigidity among finitely presented groups but not among all finitely generated groups. The authors build Γ from Seifert fibred space groups with base orbifolds drawn from explicit lists, and show that Γ×Γ is profinitely rigid in the class of finitely presented residually finite groups, while there exist infinitely many non-isomorphic finitely generated groups Λ with whose embeddings into Γ×Γ yield Grothendieck pairs. The strategy blends Galois rigidity for triangle groups, central extensions, and a pro-finite Grothendieck framework, augmented by new constructions in products of central extensions of hyperbolic groups to produce infinitely many nonisomorphic profinitely-trivial quotients. The results illuminate how finite presentability constrains profinite phenomena and provide a wide supply of Grothendieck pairs in products, contributing to a deeper understanding of profinite genus and rigidity in 3-manifold groups with arithmetic structure.

Abstract

We prove that there exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form where is a profinitely rigid 3-manifold group; we describe a family of such groups with the property that if is a finitely generated, residually finite group with then there is an embedding that induces the profinite isomorphism; in each case there are infinitely many non-isomorphic possibilities for .
Paper Structure (33 sections, 60 theorems, 40 equations)

This paper contains 33 sections, 60 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Centres
  4. Goodness
  5. Central Extensions
  6. Galois Rigidity
  7. Seifert fibred spaces
  8. Some basic facts about Seifert fibred spaces
  9. Finite extensions of Seifert fibred spaces
  10. A result of Wilkes
  11. Some calculations
  12. Profinite rigidity of certain Seifert fibred spaces
  13. Reducing to the relative case
  14. Completing the proof of profinite rigidity
  15. Reducing to Grothendieck rigidity: proof of Theorem \ref{['t3']}
  16. ...and 18 more sections

Key Result

Theorem 1.1

There exist finitely presented, residually finite groups $\Gamma$ with the following properties: If $M$ is any Seifert fibred space with base orbifold $S^2(3,3,4)$ or $S^2(3,3,6)$ or $S^2(2,5,5)$, then $\Gamma=\pi_1M$ has these properties.

Theorems & Definitions (102)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Example 2.2
  • Theorem 2.3
  • ...and 92 more