Kourovka Notebook Problem 8.44

Definitions and mathematical statement

For a prime number $p$ and an integer $r$ satisfying $r^2+1\equiv0\pmod p$, define the group \[ G_{p,r}=\left\langle a,b\ \middle|\ a^2=b^p=(ab)^3=(b^r a b^{-2r}a)^2=1\right\rangle. \] Here the presentation denotes the quotient of the free group on $a,b$ by the normal closure of the displayed relators. Since $b^p=1$, the presentation depends only on the residue class of $r$ modulo $p$.

J. Mennicke asks whether this group is infinite for all but finitely many primes $p$. The prime domain requires clarification: a permissible $r$ exists precisely when $p=2$ or $p\equiv1\pmod4$. The source also does not explicitly quantify over the possible roots $r$.

If the intended prime domain is the primes admitting such a root, and the assertion is intended to hold for every root, a precise formulation is \[ \exists\text{ a finite set }E\text{ of primes}\quad \forall p\text{ prime with }p\notin E,\quad \forall r\in{\mathbb{Z}},\qquad r^2+1\equiv0\pmod p\ \Longrightarrow\ |G_{p,r}|=\infty. \] This is an explicitly qualified reading; the supplied record does not settle whether every root or a selected root is intended.

The source reports that computer-assisted work proved $G_p$ finite for $p\leq17$, subject to the same unresolved admissibility and root conventions. It also states that a solution would have interesting topological applications, without specifying those applications. The supplied registry labels the problem open; neither that label nor the computational report is independently verified here.

Short English statement

Mennicke asks whether the displayed two-generator groups are infinite for all but finitely many eligible primes. Eligibility requires a square root of $-1$ modulo $p$; the source leaves the quantification over that root implicit. It reports computer-assisted finiteness for $p\leq17$ and potential topological applications.

Sources

• Local UnsolvedMath registry, permanent ID $700494$, exact source code catalogue entry, title Kourovka Notebook Problem 8.44. The supplied complete source record has upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82 and records status open; its update timestamp is 2026-10-16T14:00:00Z.
• J. Mennicke, Problem 8.44, The Kourovka Notebook (Unsolved Problems in Group Theory), 8th issue (1982), as attributed in the supplied record to the 21st-issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors.
• The supplied statement reports computer-assisted finiteness for $p\leq17$ and mentions topological applications. It supplies no separate citation, author, computational details, or application reference for these assertions. The cited Notebook source has not been independently consulted here.