The Diophantine problem in Thompson's group F

TL;DR

The paper establishes that the Diophantine problem is undecidable in Thompson's group $F$ by exploiting its finite commutator width and rank-$2$ abelianisation to simulate abelianisation-constrained undecidability arguments from free groups and monoids. Central to the approach is the equational definability of key subgroups and monoids in $F$, along with the definability of exponent-sum constraints, enabling the embedding of Hilbert's tenth problem into a system of equations in $F$. The authors then embed the natural numbers into the monogenic submonoid $ ext{Mon}igl\<x_0igr angle$ and define addition and multiplication via definable triples $(x_0^k,x_0^l,x_0^{k+l})$ and $(x_0^k,x_0^l,x_0^{kl})$, reducing any polynomial equation over $ obreak mathbb{N}$ to a Diophantine problem in $F$; this yields undecidability for $F$ and, via a centraliser embedding, for $T$ as well. The work also discusses the boundaries with $V$, notes on abelianisation differences, and outlines several natural questions about diagram groups and the single-equation problem. Overall, the results significantly broaden the landscape of groups with undecidable Diophantine problems and illustrate a robust method for transferring Hilbert-type undecidability through abelianisation constraints and commutator structure.

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 and Ciobanu and Garreta to show the Diophantine problem in free groups and monoids with abelianisation constraints is undecidable.

Theorems & Definitions (24)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Example 2.2
• Theorem 2.3: CannonFloydParry
• Proposition 2.5: Proof of MatucciThesis
• Definition 2.6
• Example 2.7
• Definition 2.8
• Example 2.9