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

2026-08-03

AI Summary

This study characterizes the conditions under which finitely generated groups embed into Thompson’s group $V$, along with their algebraic and geometric properties. By employing graph-theoretic methods, it establishes for the first time a necessary and sufficient condition for such an embedding: a group must admit a faithful context-free action, i.e., belong to the class CF-TR of transition groups of context-free graphs. This work bridges group theory, formal language theory, and the geometric structure of graphs, demonstrating that all known co-context-free word problem groups embed into $V$. Furthermore, it reveals that every finitely generated subgroup of $V$ is either virtually abelian or contains a non-abelian free semigroup. As applications, the results rule out the possibility of embeddings into $V$ for groups of intermediate growth, as well as for the Basilica group and the Hanoi Towers group.

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 $V$. We further study the relation between transition groups defined by graphs that are limits or covers of each others, and prove properties of transition groups of context-free graphs of polynomial growth. Finally, we prove that the Basilica and Hanoï Towers groups do not embed in $V$. This uses the geometry of Schreier graphs of the natural actions of these groups and of Thompson's $V$.

Problem

• Thompson's group V
• finitely generated subgroups
• embedding
• context-free graphs
• transition groups

Innovation

• Thompson's group V
• context-free graphs
• transition groups
• co-context-free word problem
• Schreier graphs