Kourovka Notebook Problem 17.107

Definitions and mathematical statement

Let \[ G=\mathrm{SL}_2({\mathbb{C}})=\{A\in M_2({\mathbb{C}}):\det A=1\}. \] A subgroup $H\leq G$ is free of rank two if there exist $a,b\in H$ such that $H=\langle a,b\rangle$ and every nonempty reduced word in $a^{\pm1},b^{\pm1}$ represents a nonidentity element.

Does there exist a subgroup $H\leq G$ free of rank two and an element $g\in G$ such that \[ gHg^{-1}\subsetneq H? \] Here $gHg^{-1}=\{ghg^{-1}:h\in H\}$, and the inclusion must be proper.

The supplied source reports that a free subgroup of rank six satisfying this same conjugation property has been found by D. Calegari and N. M. Dunfield. This background assertion is retained as a source report and is not independently verified here. The registry labels the rank-two question open; that label is supplied curation rather than an independently verified assessment.

Short English statement

Does $\mathrm{SL}_2({\mathbb{C}})$ contain a free group of rank two that conjugation by some element of $\mathrm{SL}_2({\mathbb{C}})$ maps onto a proper subgroup of itself? The source reports an example of rank six.

Sources

• Local UnsolvedMath registry, permanent ID $701408$, source problem code catalogue entry, title Kourovka Notebook Problem 17.107. The supplied record labels the problem open and was updated on 2026-10-16.
• The Kourovka Notebook (Unsolved Problems in Group Theory), 17th issue (2010), Problem 17.107, proposed by M. Sapir. The supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors.
• D. Calegari and N. M. Dunfield, Proc. Amer. Math. Soc. 134, no. 11 (2006), 3131--3136. Supplied reference for the reported rank-six example.