The Diophantine problem in Thompson’s group F

Luna Elliott; Alex Levine

doi:10.1090/MCOM/4113

The Diophantine problem in Thompson’s group F

Luna Elliott
, Alex Levine

School of Engineering, Mathematics and Physics

Research output: Contribution to journal › Article › peer-review

Abstract

We show that the Diophantine problem in Thompson’s group F is undecidable. Our proof uses the facts that F has finite commutator width and rank 2 abelianisation, then uses similar arguments used by Büchi and Senger [Z. Math. Logik Grundlag. Math. 34 (1988), pp. 337–342] and Ciobanu and Garreta [Int. Math. Res. Not. IMRN 5 (2024), pp. 4119–4159] to show the Diophantine problem in free groups and monoids with abelianisation constraints is undecidable.

Original language	English
Pages (from-to)	2013-2023
Number of pages	11
Journal	Mathematics of Computation
Volume	95
Issue number	360
DOIs	https://doi.org/10.1090/MCOM/4113
Publication status	Published - 9 Jun 2025

Keywords

equations in groups
Thompson’s group F

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10.1090/MCOM/4113

Cite this

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abstract = "We show that the Diophantine problem in Thompson{\textquoteright}s group F is undecidable. Our proof uses the facts that F has finite commutator width and rank 2 abelianisation, then uses similar arguments used by B{\"u}chi and Senger [Z. Math. Logik Grundlag. Math. 34 (1988), pp. 337{\textendash}342] and Ciobanu and Garreta [Int. Math. Res. Not. IMRN 5 (2024), pp. 4119{\textendash}4159] to show the Diophantine problem in free groups and monoids with abelianisation constraints is undecidable.",

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AB - We show that the Diophantine problem in Thompson’s group F is undecidable. Our proof uses the facts that F has finite commutator width and rank 2 abelianisation, then uses similar arguments used by Büchi and Senger [Z. Math. Logik Grundlag. Math. 34 (1988), pp. 337–342] and Ciobanu and Garreta [Int. Math. Res. Not. IMRN 5 (2024), pp. 4119–4159] to show the Diophantine problem in free groups and monoids with abelianisation constraints is undecidable.

KW - equations in groups

KW - Thompson’s group F

UR - https://www.scopus.com/pages/publications/105038225881

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JF - Mathematics of Computation

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ER -

The Diophantine problem in Thompson’s group F

Abstract

Keywords

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