The Diophantine problem in Thompson's group F

TL;DR

This paper proves that determining solutions to equations in Thompson's group F is undecidable, extending the understanding of Diophantine problems in algebraic structures using properties like finite commutator width.

Contribution

It establishes the undecidability of the Diophantine problem in Thompson's group F by leveraging its algebraic properties and adapting previous undecidability proofs.

Findings

Diophantine problem in F is undecidable

Uses properties like finite commutator width and rank 2 abelianisation

Adapts methods from free groups and monoids with abelian constraints

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 B\"uchi and Senger and Ciobanu and Garreta to show the Diophantine problem in free groups and monoids with abelianisation constraints is undecidable.