Kourovka Notebook Problem 21.144

Definitions and mathematical statement

Let $F$ be Thompson's group of orientation-preserving piecewise-linear homeomorphisms of $[0,1]$ under composition. Each map has finitely many breakpoints, each breakpoint is a dyadic rational $a/2^b$, and every slope is $2^k$ for some integer $k$.

The elementary amenable groups form the smallest class $\mathcal E$ containing all finite groups and all abelian groups and closed under taking subgroups, quotient groups, extensions, and directed unions. Closure under extensions means that $N\trianglelefteq G$, $N\in\mathcal E$, and $G/N\in\mathcal E$ imply $G\in\mathcal E$. A directed union is the union of a family of subgroups in which any two are contained in a third; membership of every group in that family implies membership of the union.

The conjecture is that every subgroup $H\le F$ satisfies \[ H\in\mathcal E\qquad\text{or}\qquad \exists\text{ an injective homomorphism }F\longrightarrow H. \] Equivalently, any subgroup of $F$ that is not elementary amenable must itself contain a copy of $F$.

Subgroups $H$ are arbitrary and need not be finitely generated or finitely presented. Containing a copy means abstract group containment through an injection, not merely having $F$ as a quotient or containing particular standard generators. The elementary amenable class is specified by its algebraic closure operations, so the first alternative makes a definite assertion stronger than merely belonging to some unspecified class of groups with invariant means.

Short English statement

Must every subgroup of Thompson’s group $F$ either be elementary amenable or contain a subgroup isomorphic to $F$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2653 (KOU-21.144), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.144. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.