Kourovka Notebook Problem 21.117

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 generating set $S$ of a group $G$, let $B_S(n)$ be the words of length at most $n$ in $S\cup S^{-1}$ and put $\lambda(G,S)=\lim_{n\to\infty}|B_S(n)|^{1/n}$. Exponential growth means $\lambda(G,S)>1$; uniform exponential growth means $\inf_S\lambda(G,S)>1$, with the infimum over all finite generating sets. An infinite group is simple if its only normal subgroups are $1$ and itself. It is just-infinite if every nontrivial normal subgroup has finite index, and hereditarily just-infinite if every finite-index subgroup is just-infinite.

Statement recorded in the supplied source.

Does there exist (a) a finitely generated infinite simple group, and (b) a finitely generated hereditarily just-infinite group, of exponential growth but without uniform exponential growth? These are two separate existence questions. [S1][S2]

Short English statement

Ask whether simple groups and hereditarily just-infinite groups can have exponential growth whose rate becomes arbitrarily close to one as the generators change.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2626 (KOU-21.117). 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 21.117 and its remarks. https://arxiv.org/abs/1401.0300v48.