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Some remarks on Grothendieck pairs

Andrei Jaikin-Zapirain, Alexander Lubotzky

TL;DR

This work revisits Grothendieck pairs and analyzes the interplay between profinite rigidity and left/right Grothendieck rigidity (LGR/RGR). It re-proves Grothendieck's rigidity framework, identifies broad classes of groups with LGR/RGR properties, and proves that ascending $HNN$-extensions of finitely generated free groups are $RGR$, highlighting a new, robust rigidity phenomenon. The authors separate Grothendieck rigidity from profinite invariants by presenting explicit counterexamples involving Bohr and proalgebraic completions, and they explore how these notions interact with commensurability and residual properties. They also outline numerous open questions about rigidity across Bohr, proalgebraic, and profinite settings, providing a roadmap for further study of the rigidity landscape in group theory.

Abstract

We revisit the paper of Alexander Grothendiek where he introduced Grothendieck pairs and discuss the relation between profinite rigidity and left/right Grothendieck rigidity. We also show that various groups are left and/or right Grothendieck rigid and, in particular, all ascending HNN extensiona of finitely generated free groups are right Grothendieck rigid. Along the way we present a number of questions and suggestions for further research.

Some remarks on Grothendieck pairs

TL;DR

This work revisits Grothendieck pairs and analyzes the interplay between profinite rigidity and left/right Grothendieck rigidity (LGR/RGR). It re-proves Grothendieck's rigidity framework, identifies broad classes of groups with LGR/RGR properties, and proves that ascending -extensions of finitely generated free groups are , highlighting a new, robust rigidity phenomenon. The authors separate Grothendieck rigidity from profinite invariants by presenting explicit counterexamples involving Bohr and proalgebraic completions, and they explore how these notions interact with commensurability and residual properties. They also outline numerous open questions about rigidity across Bohr, proalgebraic, and profinite settings, providing a roadmap for further study of the rigidity landscape in group theory.

Abstract

We revisit the paper of Alexander Grothendiek where he introduced Grothendieck pairs and discuss the relation between profinite rigidity and left/right Grothendieck rigidity. We also show that various groups are left and/or right Grothendieck rigid and, in particular, all ascending HNN extensiona of finitely generated free groups are right Grothendieck rigid. Along the way we present a number of questions and suggestions for further research.
Paper Structure (7 sections, 19 theorems, 42 equations)

This paper contains 7 sections, 19 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Commutative algebra preliminaries
  3. Grothendieck's theorem
  4. Grothendieck left/right rigid groups
  5. Ascending $HNN$-extensions of finitely generated free groups
  6. Grothendieck properties versus profinite properties
  7. Additional questions

Key Result

Theorem 1.1

The following holds.

Theorems & Definitions (38)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 2.1
  • Corollary 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 28 more