Kourovka Notebook Problem 16.89

Definitions and mathematical statement

Let $X$ be an infinite set and let $F(X)$ be the free group with free basis $X$. Write $G_X=\operatorname{Aut}(F(X))$ for its group of automorphisms. Interpret finite width as the following property: for every generating subset $S\subseteq G_X$, there is an integer $n\geq 1$ such that every element of $G_X$ is a product of at most $n$ elements of $S\cup S^{-1}$, where $S^{-1}=\{s^{-1}:s\in S\}$. Equivalently, \[ \forall S\subseteq G_X,\qquad \langle S\rangle=G_X\ \Longrightarrow\ \exists n\geq 1:\quad G_X=(S\cup S^{-1}\cup\{1\})^n. \] The bound may depend on $X$ and $S$.

Question (catalogue entry). Does $\operatorname{Aut}(F(X))$ have finite width in this sense for every infinite set $X$?

The supplied statement reports an affirmative answer when $|X|=\aleph_0$; consequently, the remaining question concerns uncountable free bases. This affirmative countable-rank assertion is supplied background, not an independently verified result here. The source record does not define finite width; the formulation above explicitly adopts its usual interpretation in terms of every generating subset, and this interpretation requires source confirmation.

Short English statement

Does the automorphism group of every free group of infinite rank have bounded word length with respect to each of its generating subsets, with the bound allowed to depend on the subset and the group? The supplied record reports that the countable-rank case has an affirmative answer and labels the overall problem open. Its undefined term finite width requires confirmation against the source.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $701280$, exact source code catalogue entry, title Kourovka Notebook Problem 16.89. The record supplies the status open; this status is not independently verified here.
• The Kourovka Notebook (Unsolved Problems in Group Theory), Problem 16.89, originating in the 16th issue (2006), as identified in the supplied record with the 21st issue edition (2026), arXiv:1401.0300v48. The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. No source consultation is claimed.
• Attribution: the record's proposer field and background name V. Tolstykh, with proposed year 2006; the statement itself begins with the attribution "(G. M. Bergman)." Both attributions are retained without resolving their respective roles.