Kourovka Notebook Problem 20.59

Definitions and mathematical statement

A group is finitely generated if it is generated by a finite subset. A group $G$ is virtually free if it contains a free subgroup $F\leq G$ of finite index, meaning $[G:F]<\infty$. A subgroup $H\leq K$ is a retract of $K$ if there is a group homomorphism $r:K\to H$ satisfying $r(h)=h$ for every $h\in H$. A subgroup $H\leq G$ is a virtual retract of $G$ if there exists a subgroup $K$ such that $H\leq K\leq G$, $[G:K]<\infty$, and $H$ is a retract of $K$.

The question proposed by A. Minasyan in Kourovka Notebook Problem 20.59 (2022) is: for every finitely generated virtually free group $G$ and every finitely generated subgroup $H\leq G$, do there exist a subgroup $K$ and a group homomorphism $r$ such that \[ H\leq K\leq G,\qquad [G:K]<\infty,\qquad r:K\to H,\qquad \forall h\in H,\ r(h)=h? \] The supplied formulation imposes no normality requirement on $K$.

Short English statement

Must every finitely generated subgroup of a finitely generated virtually free group be a retract of some finite-index subgroup containing it?

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $701717$, exact source code catalogue entry, title Kourovka Notebook Problem 20.59. The supplied registry labels the problem open; this label has not been independently verified.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 20th issue (2022), Problem 20.59, proposed by A. Minasyan. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The registry describes the Issues 1--21 collection as edited by E. I. Khukhro and V. D. Mazurov.
• A. Minasyan, Int. Math. Res. Notes, 2021, no. 17 (2021), 13434--13477. The supplied statement cites this reference for motivation and partial results; no specific partial-result statement is supplied.