Kourovka Notebook Problem 21.44

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.

Let $A_5$ be the alternating group acting on five letters. Let $W_n$ be the automorphism group of the rooted five-ary tree of height $n$ whose permutation of the five children at every vertex belongs to $A_5$. Thus $W_1=A_5$ and $W_{n+1}=A_5^{5^n}\rtimes W_n$, the natural iterated permutational wreath product. Truncation gives $W_{n+1}\to W_n$; define the profinite group $W=\varprojlim W_n$ with its inverse-limit topology. A subgroup $\Lambda\leq W$ is dense if its projection onto each $W_n$ is surjective. For a finite symmetric generating set $S$ of $\Lambda$, let $b_S(m)$ count elements represented by words of length at most $m$. Subexponential growth means $\lim_{m\to\infty}b_S(m)^{1/m}=1$, independently of the chosen finite generating set.

Statement recorded in the supplied source.

Does $W$ contain a finitely generated dense subgroup $\Lambda$ of subexponential growth? There is no requirement that $\Lambda$ be closed in $W$. [S2]

Short English statement

Does the infinite iterated alternating-group wreath product have a finitely generated dense subgroup with subexponential word growth?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2553 (KOU-21.44). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 S. Eberhard, Problem 21.44, in E. I. Khukhro and V. D. Mazurov (eds.), The Kourovka Notebook: Unsolved Problems in Group Theory, 21st edition, arXiv:1401.0300v48 (6 October 2026). This version records an affirmative answer; the displayed question is the historically registered target. https://arxiv.org/abs/1401.0300v48.