Questions in Geometric Group Theory — Q 8.9

Definitions and mathematical statement

Let $F$ be a free group of finite rank and let $N\trianglelefteq F$ be a normal subgroup normally generated by finitely many elements. Put $G=F/N$, a finitely presented group. Define the normal generation number \[ d_F(N)=\min\{|R|:R\subseteq N, \langle\!\langle R\rangle\!\rangle_F=N\}. \] The relation module is the abelian group $N/[N,N]$, equipped with a left $\mathbb ZG$-module structure by conjugation: if $g$ is represented by $f\in F$, then $g\cdot[n]=[fnf^{-1}]$. This is well-defined because conjugation by an element of $N$ is trivial on its abelianization.

Let $d_G(N/[N,N])$ be the smallest number of generators of this module over $\mathbb ZG$. Does there exist a pair $(F,N)$ as above for which \[ d_G(N/[N,N])<d_F(N)<\infty? \] Images of normal generators always generate the relation module, so the reverse weak inequality holds automatically. The question asks for strictness: can the abelianized relations be generated with fewer elements than the normal subgroup itself?

Both minima refer to this fixed free presentation, not to minimization over all presentations of $G$. The module generators may be arbitrary elements of the relation module and are not required to be a subset of a displayed relator list. Counting ordinary abelian generators instead would ignore the conjugation action and give a different problem.

Short English statement

Can the relation module of a finite presentation need fewer module generators than its relation subgroup needs normal generators?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100809 (AMR-010-0809), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 8.9 (PDF page 17). Original source indexed by AMR-010-0809. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.