AI summaryⓘ
The authors show that whether a small, finitely generated group can fit into Thompson's group V depends on a certain way the group acts, called a faithful context-free action. Using this, they confirm that many groups known to have a certain easy-to-check problem (co-context-free Word Problem) do fit inside V. They also find that every small group inside V is either almost like a group where everything commutes or contains a kind of complicated part called a free non-abelian semigroup, which means groups that grow in between these types cannot fit inside V. Lastly, they prove that two particular groups called Basilica and Hanoï Towers do not fit into V by analyzing the shapes of their action diagrams. The authors connect graph theory, group actions, and algebraic properties in their work.
Thompson's group Vfinitely generated groupfaithful context-free actionco-context-free Word Problemfree non-abelian semigroupgroups of intermediate growthSchreier graphtransition groupscontext-free graphsembedding of groups
Authors
Corentin Bodart, Daniele D'Angeli, Davide Perego, Emanuele Rodaro
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 $V$.