Kourovka Notebook Problem 21.145

Definitions and mathematical statement

Thompson's group $F$ consists of orientation-preserving piecewise-linear homeomorphisms of $[0,1]$ with finitely many dyadic rational breakpoints and slopes in $\{2^k:k\in\mathbb Z\}$, with composition as its operation. It is finitely generated; give it a word metric from a finite generating set. Give the infinite cyclic group $\mathbb Z$ its metric $|m-n|$, and each direct product a word metric, equivalently the sum of the factor word metrics.

A map $f:X\to Y$ between metric spaces is a quasi-isometry if there are constants $\lambda\ge1$, $C\ge0$ such that \[ \lambda^{-1}d_X(x,x')-C\le d_Y(f(x),f(x')) \le\lambda d_X(x,x')+C \] for every $x,x'\in X$, and every point of $Y$ is at distance at most $C$ from $f(X)$. Existence of a quasi-isometry between finitely generated groups does not depend on the finite generating sets used.

The two questions are: (a) Is $F$ quasi-isometric to $F\times\mathbb Z$? (b) Is $F$ quasi-isometric to $F\times F$?

Each part asks for a map satisfying both the distance inequalities and coarse surjectivity; the maps and constants may differ between the two parts. A quasi-isometry is not required to be a homomorphism, injective, or continuous in the interval-action model. These are therefore comparisons of the large-scale word-metric spaces of the indicated groups.

Short English statement

Is Thompson’s group $F$ quasi-isometric to its product with $\mathbb Z$, or to its square?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2654 (KOU-21.145), 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 21.145. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.