Kourovka Notebook Problem 21.143

Definitions and mathematical statement

Thompson's group $F$ is the group, under composition, of orientation-preserving piecewise-linear homeomorphisms of $[0,1]$ with finitely many breakpoints, all breakpoints dyadic rational, and every slope a power of $2$ with integer exponent. A dyadic rational is a number $a/2^b$ with $a\in\mathbb Z$ and $b\ge0$.

For a finitely generated group $G$, take a finite symmetric generating set $S$ and its word metric $d_S$. Let $S^*$ be the set of finite words in $S$, and write $\overline u$ for the group element represented by a word $u$. A language $L\subseteq S^*$ is regular if some finite-state automaton accepts exactly its words.

An automatic structure can be specified by such a regular language $L$ representing every element of $G$, together with a constant $K\ge0$ satisfying the synchronous fellow-traveler condition. For every $u,v\in L$ with $d_S(\overline u,\overline v)\le1$, and every integer $t\ge0$, require \[ d_S\left(\overline{u_{\le t}},\overline{v_{\le t}}\right)\le K, \] where $u_{\le t}$ is the prefix of length $\min(t,|u|)$, so a path remains at its endpoint after its word ends. A group is automatic if such $S,L,K$ exist.

Is Thompson's group $F$ automatic in this sense? The requested language need not consist of geodesics or be the usual piecewise-linear normal form. The same finite automaton and fellow-traveler bound must work for all elements and all neighboring represented endpoints.

Short English statement

Does Thompson’s group $F$ admit a regular language of representatives with a uniform synchronous fellow-traveler bound?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2652 (KOU-21.143), 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 21.143. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.