Kourovka Notebook Problem 21.79

Definitions and mathematical statement

For a group $G$ and finite generating set $S$, let $|g|_S$ be the least length of a word in $S\cup S^{-1}$ representing $g$, with $|1|_S=0$. An element $g\in G$ is distorted if \[ \lim_{n\to\infty}\frac{|g^n|_S}{n}=0. \] The limit exists by subadditivity of $n\mapsto|g^n|_S$, and the zero-limit condition is independent of the chosen finite generating set: any two such word lengths bound one another up to constant factors.

A left order on $G$ is a strict total order $<$ satisfying $x<y\Rightarrow ax<ay$ for every $a,x,y\in G$. The group is left-orderable if such an order exists. Every nonidentity element of a left-orderable group has infinite order, so distortion here concerns powers of infinite-order elements.

Does there exist a nontrivial finitely generated left-orderable group $G$ such that every $g\in G\setminus\{1\}$ is distorted? In quantified form, for one finite generating set $S$, every $g\ne1$ and every real $\varepsilon>0$ must admit $N=N(g,\varepsilon)$ such that $|g^n|_S\le\varepsilon n$ for all integers $n\ge N$. Neither $N$ nor the rate of convergence is required to be uniform in $g$. Nontriviality excludes the vacuous one-element group from the existence question.

Short English statement

Find a nontrivial finitely generated left-orderable group in which the powers of every nonidentity element have sublinear word length.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2588 (KOU-21.79), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.79. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.