Kourovka Notebook Problem 20.44

Definitions and mathematical statement

The supplied statement refers to Problem 17.57 for the definition of $\mathrm{CT}({\mathbb{Z}})$, but that definition is not included. Consequently, the ambient group and its action cannot be specified completely from this record. The statement presupposes an action of $\mathrm{CT}({\mathbb{Z}})$ on ${\mathbb{Z}}$.

Relative to that intended action, for a subgroup $H\leq \mathrm{CT}({\mathbb{Z}})$ define \[ H\cdot n=\{h\cdot n:h\in H\},\qquad \mathcal O_H=\{H\cdot n:n\in{\mathbb{Z}}\}. \] The subgroup $H$ is finitely generated if $H=\langle F\rangle$ for some finite subset $F\subseteq H$. A set of orbit representatives is a subset $R\subseteq{\mathbb{Z}}$ satisfying \[ \forall O\in\mathcal O_H,\qquad |R\cap O|=1. \] The source asks whether every finitely generated subgroup $H\leq\mathrm{CT}({\mathbb{Z}})$ satisfies \[ |\mathcal O_H|<\infty \quad\text{or}\quad \exists R\subseteq{\mathbb{Z}}\;\left[ (\forall O\in\mathcal O_H,\ |R\cap O|=1) \ \land\ R\text{ has positive density}\right]. \] Here "positive density" retains the source wording: the record does not specify the density convention on ${\mathbb{Z}}$. This is therefore a conditional transcription of the question, pending the definition in Problem 17.57 and clarification of density, rather than a fully specified conjecture.

Short English statement

Must every finitely generated subgroup of $\mathrm{CT}({\mathbb{Z}})$ either have finitely many orbits on ${\mathbb{Z}}$ or admit a set containing exactly one representative from each orbit and having positive density? The supplied record omits the definition of $\mathrm{CT}({\mathbb{Z}})$ and the intended density convention.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $701702$, exact source problem code catalogue entry, title "Kourovka Notebook Problem 20.44." The registry labels the problem open; this is supplied metadata, not an independently verified status assessment.
• Attributed source: The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 20.44, proposed by S. Kohl in 2022. The supplied background names the 20th issue (2022) and the 21st issue edition (2026), with identifier arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The statement explicitly refers to Problem 17.57 for the definition of $\mathrm{CT}({\mathbb{Z}})$. That referenced definition is absent from the supplied record. The cited source and cross-reference have not been independently consulted or verified.