The Diophantine problem in Thompson’s group 𝐹

We show that the Diophantine problem in Thompson’s group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is undecidable. Our proof uses the facts that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has finite commutator width and rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.