Kourovka Notebook Problem 15.84

Definitions and mathematical statement

For scalars $\alpha,\beta,\gamma$, define \[ A_\alpha=\begin{pmatrix}1&\alpha\\0&1\end{pmatrix},\qquad B_\beta=\begin{pmatrix}1&0\\\beta&1\end{pmatrix},\qquad C_\gamma=\begin{pmatrix}1-\gamma&-\gamma\\\gamma&1+\gamma\end{pmatrix}. \] Each matrix has determinant one. Let $F_3=\langle x,y,z\rangle$ denote the free group on three specified generators.

The supplied record attributes to Yu. I. Merzlyakov the result that, for $\alpha,\beta,\gamma\in{\mathbb{C}}$ satisfying $|\alpha|,|\beta|,|\gamma|\geq 3$, these matrices generate a free group of rank three. This background result is reported from the supplied source and has not been independently verified here.

The question is whether there exist $\alpha,\beta,\gamma\in{\mathbb{Q}}$ satisfying \[ |\alpha|<3,\qquad |\beta|<3,\qquad |\gamma|<3 \] such that the homomorphism \[ \varphi_{\alpha,\beta,\gamma}:F_3\longrightarrow\operatorname{SL}_2({\mathbb{Q}}),\qquad x\longmapsto A_\alpha,\quad y\longmapsto B_\beta,\quad z\longmapsto C_\gamma \] is injective. Equivalently, every nonidentity freely reduced word in $x,y,z$ evaluates to a matrix different from $I_2$. Thus the three displayed matrices must freely generate their subgroup, which has rank three.

The mathematical question is explicit in the supplied record; the unresolved source qualification concerns the truncated spelling of the proposer's name.

Short English statement

Can three rational parameters, each strictly between $-3$ and $3$, make the displayed matrices freely generate a group of rank three? The supplied background states that complex parameters of absolute value at least $3$ have this property.

Sources

• Local UnsolvedMath registry, permanent ID $701172$, exact source problem code catalogue entry, title Kourovka Notebook Problem 15.84; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), problem 15.84, originating in the 15th issue (2002), as identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48, https://arxiv.org/abs/1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. The problem is attributed to Yu. V. Sosnovski; the supplied terminal diacritic encoding is truncated, so the complete spelling remains unresolved.
• Yu. I. Merzlyakov, Sov. Math. Dokl. 19 (1978), 64--68, cited in the supplied statement for the complex-parameter background result. No article title is supplied.
• The registry labels this problem open. This is a supplied status label, not an independently verified literature assessment. The cited sources have not been independently consulted for this draft.