Some Questions — Question 5

Definitions and mathematical statement

Let $C_2$ be the cyclic group of order two. Define finite groups recursively by $W_1=C_2$ and \[ W_{n+1}=(W_n\times W_n)\rtimes C_2, \] where the nonidentity element of $C_2$ interchanges the two factors. Thus $W_n$ is the full automorphism group of the rooted binary tree of depth $n$, and specifies the permutational iterated wreath product convention unambiguously.

A nontrivial group word is a nonidentity element $w$ of a free group on finitely many variables. Its length $|w|$ is the number of letters in its freely reduced expression, counting inverses as single letters. The word is a law for $W_n$ if every assignment of its variables to elements of $W_n$ evaluates to the identity.

Define \[ \ell(n)=\min\{|w|:w\ne1\text{ is a law for }W_n\}. \] Determine the asymptotic growth of $\ell(n)$ as $n\to\infty$, with matching upper and lower estimates where possible. The number of variables may vary with the word; the minimization is not restricted to power words or a particular presentation. The power word $x^{2^n}$ is a law and supplies an elementary benchmark, but the problem is whether much shorter identities involving several variables can exist. A word becoming trivial only for selected assignments is not a law. The group parameter is the tree depth, rather than the logarithm of the group order.

Short English statement

How long is the shortest nontrivial identity in the automorphism group of a finite rooted binary tree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200005 (AMR-011-0005), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Abert - Some questions (pdf) (2010), Question 5. Original source indexed by AMR-011-0005. https://www.renyi.hu/~abert/questions.pdf.