Kourovka Notebook Problem 21.16

Definitions and mathematical statement

For a group $G$ and a subset $X\subseteq G$ generating $G$ as a group, let $\ell_X^{\pm}(g)$ be the least number of letters from $X\cup X^{-1}$ in a word equal to $g$. The empty word has length zero and represents $1$. The group width with respect to $X$ is $\sup_{g\in G}\ell_X^{\pm}(g)$, possibly infinite. Say $G$ has finite group width if this supremum is finite for every group-generating set $X$; the bound may depend on $X$.

Independently, $Y\subseteq G$ generates $G$ as a monoid if every element is a finite product of members of $Y$, without automatically adjoining inverses. Let $\ell_Y^+(g)$ be the least such product length. Finite monoid width means $\sup_g\ell_Y^+(g)<\infty$ for every monoid-generating set $Y$.

Does there exist a group $G$ with finite group width but not finite monoid width? A positive answer requires both \[ \forall X\text{ group-generating},\quad\sup_g\ell_X^{\pm}(g)<\infty, \qquad \exists Y\text{ monoid-generating},\quad\sup_g\ell_Y^+(g)=\infty. \] Generating sets may be infinite. There is no common width bound across all generating sets in either definition. Since a finite group has finite width for each generating set in either sense, a witness must be infinite. The two conditions concern the same underlying group; the distinction is whether formal inverse letters are freely available.

Short English statement

Can a group have bounded word length for each group-generating set but unbounded positive-word length for some monoid-generating set?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2525 (KOU-21.16), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.16. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.