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A graph-theoretical characterisation of subgroups of Thompson's group V · arXivDesk

2608.02111Aug 3, 2026Preliminary version. 44 pages, 19 figures

A graph-theoretical characterisation of subgroups of Thompson's group $V$

Corentin Bodart, Daniele D'Angeli, Davide Perego, Emanuele Rodaro

Group Theory Formal Languages and Automata Theory Combinatorics

Abstract

We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group $V$ : a finitely generated group embeds in $V$ if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors. Using this characterisation, we prove results in different directions: - All known examples of groups with co-context-free Word Problem do embed in $V$ , providing evidence towards Lehnert's conjecture. - Each finitely generated subgroup of $V$ is either virtually abelian, or contains a free non-abelian semigroup. It follows that groups of intermediate growth do not embed in Thompson's $V$

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We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group V: a finitely generated group embeds in V if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors. Using this characterisation, we prove results in different directions: - All known examples of groups with co-context-free Word Problem do embed in V, providing evidence towards Lehnert's conjecture.

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The paper addresses: A graph-theoretical characterisation of subgroups of Thompson's group V.
We prove a graph-theoretical characterisation of finitely generated subgroups of Thompson's group V: a finitely generated group embeds in V if and only if it admits a faithful context-free action, or equivalently if it belongs to the class CF-TR of transition groups of context-free graphs recently introduced by Matucci and the three last authors.