Kourovka Notebook Problem 21.126

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a finite graph $\Lambda$, its right-angled Artin group has generators $v\in V(\Lambda)$ and relations $[v,w]=1$ for the edges. For a finite presentation of a group $H$, its Dehn function $\delta_H(n)$ is the greatest minimal number of relators needed to fill a null word of length at most $n$. Functions are compared up to the usual equivalence generated by $f\preccurlyeq g$ when $f(n)\leq Cg(Cn+C)+Cn+C$ for some $C>0$. Here super-exponential means faster than every exponential, and sub-exponential but nonpolynomial means exponential rate zero while no polynomial upper bound exists, in this coarse sense.

Statement recorded in the supplied source.

Do finitely presented subgroups $H$ of right-angled Artin groups exist whose Dehn functions are (a) super-exponential, or (b) sub-exponential and not polynomially bounded? [S1][S2]

Short English statement

Find finitely presented subgroups of right-angled Artin groups with either faster-than-exponential filling or slower-than-exponential filling that still exceeds every polynomial.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2635 (KOU-21.126). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 E. I. Khukhro and V. D. Mazurov (editors), The Kourovka Notebook, Unsolved Problems in Group Theory, No.21, version48 (6 October2026), Problem 21.126 and its remarks. https://arxiv.org/abs/1401.0300v48.