An exotic presentation of ℤ × ℤ and the Andrews–Curtis conjecture

Barmak, Jonathan Ariel

doi:10.2140/agt.2025.25.345

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An exotic presentation of $\mathbb{Z} \times \mathbb{Z}$ and the Andrews–Curtis conjecture

Jonathan Ariel Barmak

Algebraic & Geometric Topology 25 (2025) 345–355

DOI: 10.2140/agt.2025.25.345

Abstract

We prove that the presentations $⟨ x, y ∣ [x, y], 1 ⟩$ and $⟨ x, y ∣ {[x, [x, y^{- 1}]]}^{2} y [y^{- 1}, x] y^{- 1}, [x, [[y^{- 1}, x], x]] ⟩$ are not $Q^{*}$ -equivalent even though their standard complexes have the same simple homotopy type.

Keywords

group presentations, $Q^*$-transformations, generalized Andrews–Curtis conjecture, simple homotopy type

Mathematical Subject Classification

Primary: 20F05, 20F65, 57M05, 57Q10

References

Publication

Received: 24 July 2023

Revised: 5 December 2023

Accepted: 28 January 2024

Published: 5 March 2025

Authors

Jonathan Ariel Barmak
Departamento de Matemática Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires Buenos Aires Argentina	Instituto de Investigaciones Matemáticas Luis A Santaló (IMAS) CONICET-Universidad de Buenos Aires Buenos Aires Argentina

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Volume 25, issue 1 (2025)