Kourovka Notebook Problem 15.83

Definitions and mathematical statement

For $\alpha\in{\mathbb{Q}}$, define \[ A_\alpha=\begin{pmatrix}1&\alpha\\0&1\end{pmatrix},\qquad B_\alpha=\begin{pmatrix}1&0\\\alpha&1\end{pmatrix} \in\operatorname{SL}_2({\mathbb{Q}}), \] and let $G_\alpha=\langle A_\alpha,B_\alpha\rangle$. Let $F(a,b)$ be the free group on two generators, and let \[ \varphi_\alpha:F(a,b)\longrightarrow G_\alpha, \qquad a\longmapsto A_\alpha,\quad b\longmapsto B_\alpha \] be the homomorphism determined by these images.

Under the intended interpretation that the displayed matrices freely generate a group of rank two, the question is whether \[ \exists\alpha\in{\mathbb{Q}}\quad\left(|\alpha|<2\ \text{ and }\ \ker\varphi_\alpha=\{1\}\right). \] Equivalently, every nonempty reduced word in $a^{\pm1},b^{\pm1}$ must evaluate at $A_\alpha,B_\alpha$ to a matrix different from $I_2$.

The source says only "generate a free group" and does not explicitly specify rank two or exclude $\alpha=0$. This qualification is unresolved: at $\alpha=0$, the generated group is trivial, which is a free group of rank zero under the usual convention. The displayed injectivity formulation makes the intended nonabelian reading explicit, but that reading requires confirmation.

Short English statement

Is there a rational parameter $\alpha$ with $|\alpha|<2$ for which the two displayed unipotent matrices freely generate a free group of rank two? The source does not explicitly specify the rank; allowing the trivial free group would admit $\alpha=0$.

Sources

• Local UnsolvedMath registry, permanent ID $701171$, exact source problem code $\text{catalogue entry}$, titled Kourovka Notebook Problem 15.83.
• Supplied reference: The Kourovka Notebook (Unsolved Problems in Group Theory), 15th issue (2002), problem 15.83; also identified in the record with the 21st issue edition (2026), arXiv:1401.0300v48. These bibliographic details are supplied source data and have not been independently verified.
• The statement attributes the question parenthetically to Yu. I. Merzlyakov. The background and proposer field instead name Yu. V. Sosnovski followed by an incomplete LaTeX accent sequence; the supplied proposed year is 2002. Both attributions are retained without identifying their respective roles.
• The registry labels the problem open. This is a supplied status label, not an independently verified assessment of the literature.