Questions in Geometric Group Theory — Q 1.17

Definitions and mathematical statement

For a finitely generated group $G$, let $d(G)$ be the minimum size of a generating set and let $G^n$ be the direct product of $n$ copies of $G$. Projection shows that $d(G^n)$ is nondecreasing in $n$. Thus failure of divergence is equivalent to a uniform finite bound: \[ \sup_{n\geq1}d(G^n)<\infty. \] Find a nontrivial finitely generated group with this property in a geometrically controlled class, especially among CAT(0) groups or automatic groups.

A CAT(0) group admits a proper cocompact isometric action on a proper geodesic metric space whose geodesic triangles are no thicker than their Euclidean comparison triangles. An automatic group has, for a finite generating set, a regular language of representative words with the synchronous fellow-traveller property: representative paths whose endpoints differ by one generator stay a uniformly bounded distance apart at equal integer times.

The group $G$ must be fixed while $n$ varies; choosing different groups for different powers does not answer the question. The trivial group is excluded because all its powers require no generators. The question seeks an example in either indicated class, rather than asserting these two classes coincide. The bounded generating sets for $G^n$ may depend on $n$ and may mix coordinates; they need not be unions of generating sets for the individual factors.

Short English statement

Can a nontrivial CAT(0) or automatic group have direct powers needing only a bounded number of generators?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100117 (AMR-010-0117), 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 1.17 (PDF page 4). Original source indexed by AMR-010-0117. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.