Kourovka Notebook Problem 16.84

Definitions and mathematical statement

For an integer $n\geq 4$, let $\mathfrak B_n$ be the braid group on $n$ strands, with presentation \[ \mathfrak B_n=\left\langle \sigma_1,\ldots,\sigma_{n-1}\ \middle|\ \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\ (1\leq i\leq n-2),\quad \sigma_i\sigma_j=\sigma_j\sigma_i\ (|i-j|\geq2)\right\rangle. \] Under the usual finite-branching interpretation of a regular rooted tree, choose an integer $d\geq2$ and an alphabet $X$ with $|X|=d$. The tree $X^*$ has all finite words over $X$ as vertices, the empty word as root, and edges joining $v$ to $vx$ for $x\in X$. Its automorphisms preserve the root and adjacency. For $g\in\operatorname{Aut}(X^*)$ and $v\in X^*$, the section $g|_v\in\operatorname{Aut}(X^*)$ is determined by \[ g(vw)=g(v)\,g|_v(w)\qquad(w\in X^*). \] An automorphism $g$ is finite-state if $\{g|_v:v\in X^*\}$ is finite. An action $\rho:\mathfrak B_n\to\operatorname{Aut}(X^*)$ is faithful if $\ker\rho=\{1\}$, and is by finite-state automorphisms if every $\rho(b)$, for $b\in\mathfrak B_n$, is finite-state.

The question is whether, for every integer $n\geq4$, there exist an integer $d\geq2$, an alphabet $X$ of cardinality $d$, and an injective homomorphism \[ \rho_n:\mathfrak B_n\longrightarrow\operatorname{Aut}(X^*) \] such that \[ \forall b\in\mathfrak B_n,\qquad \left|\{\rho_n(b)|_v:v\in X^*\}\right|<\infty. \] Here the tree degree and action may depend on $n$; no common bound on the number of states is imposed. The source does not explicitly specify the tree degree or finite branching, so the finite-alphabet convention used above requires confirmation against its cited definitions. The source also states that such an action is known for $\mathfrak B_3$; this is preserved as supplied background, not as an independently verified result.

Short English statement

For each $n\geq4$, can the braid group on $n$ strands act faithfully on a regular rooted tree so that each group element acts with only finitely many distinct sections? The formulation uses the conventional finite-branching interpretation; the source leaves the tree degree unspecified. It reports that the corresponding action exists for $\mathfrak B_3$.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701275$, exact source code catalogue entry, title Kourovka Notebook Problem 16.84. The registry labels the problem open; that label is supplied metadata, not an independently verified literature assessment. The record gives proposal year 2006 and upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 16.84, 16th issue (2006); the supplied background also identifies the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These publication details are supplied attribution and have not been independently checked.
• The registry attributes the problem to V. I. Sushchanskiĭ, but its proposer field contains a truncated TeX rendering of the surname. The complete surname rendering used here occurs in the reference supplied with the statement.
• Reference supplied for definitions: R. I. Grigorchuk, V. V. Nekrashevich, V. I. Sushchanskiĭ, Proc. Steklov Inst. Math., 2000, no. 4 (231), 128--203. No article title or direct link is supplied, and the reference has not been consulted here.