Kourovka Notebook Problem 21.105

Definitions and mathematical statement

A group word in $r$ variables is an element $w\in F_r$, the free group on symbols $x_1,\ldots,x_r$. Substituting $g_1,\ldots,g_r$ in a group $G$ gives an element $w(g_1,\ldots,g_r)$. Define the set of word values and the verbal subgroup by \[ G_w=\{w(g_1,\ldots,g_r):g_i\in G\},\qquad w(G)=\langle G_w\rangle. \] The subgroup brackets mean all finite products of elements of $G_w$ and their inverses.

A group is residually finite if for every nonidentity element $g$ there is a homomorphism $\theta:G\to Q$ to a finite group such that $\theta(g)\ne1$. Equivalently, the intersection of all finite-index normal subgroups of $G$ is trivial. The group itself may be infinite and need not be finitely generated.

Is every word concise in this class? In full quantifiers, for every integer $r\ge1$, every word $w\in F_r$, and every residually finite group $G$, does \[ |G_w|<\infty\quad\Longrightarrow\quad |w(G)|<\infty \] hold? Finiteness of the value set is the hypothesis, whereas finiteness of its generated subgroup is the desired conclusion. A finite set of arbitrary group elements can generate an infinite subgroup, so their origin as all values of one fixed word matters. No universal numerical bound on $|w(G)|$ is requested here.

Short English statement

In any residually finite group, must finitely many values of a fixed group word generate a finite subgroup?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2614 (KOU-21.105), 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 21.105. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.