Questions in Geometric Group Theory — Q 1.16

Definitions and mathematical statement

For a finitely generated group $H$, let $d(H)$ be the minimum cardinality of a generating set. Generators here generate as an abstract group, without taking any topological closure. For a fixed group $G$ and an integer $n\geq1$, let $G^n$ be its direct product with coordinatewise multiplication.

Assume that $G$ is nontrivial and word-hyperbolic: a Cayley graph for a finite generating set has uniformly thin geodesic triangles. The question asks whether \[ \lim_{n\to\infty}d(G^n)=\infty. \] Explicitly, for every integer $R\geq0$, must there exist $N=N(G,R)$ such that $d(G^n)>R$ for every $n\geq N$? The group remains fixed while the number of factors grows.

The projection $G^{n+1}\to G^n$ shows that these ranks form a nondecreasing sequence. Thus the assertion is equivalent to their being unbounded, and a counterexample would have one finite bound for all powers. No particular rate of divergence or bound uniform over hyperbolic groups is requested. The trivial group must be excluded: all its direct powers have rank zero. This exclusion is also consistent with the source's accompanying discussion of nontrivial finite groups. The problem is about independent direct-product coordinates, not the ranks of finite-index subgroups or of successive quotients of $G$.

Short English statement

For a fixed nontrivial hyperbolic group, must the number of generators of its direct powers grow without bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100116 (AMR-010-0116), 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 1.16 (PDF page 4). Original source indexed by AMR-010-0116. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.