Action graphs, semiconjugacy, and non-embedding in Thompson's group $V$

TL;DR

This paper investigates subgroups of Thompson's group V, showing their action graphs are tree-like, characterizing embeddings of certain groups into V, and proving some groups cannot embed into V, including F_{2,3}.

Contribution

It establishes new structural properties of subgroups of V and provides criteria for non-embedding of specific groups like F_{2,3} into V.

Findings

Action graphs of finitely generated subgroups of V are quasi-isometric to trees.

Embeddings of certain homeomorphism groups into V are semiconjugate to standard actions.

The Stein group F_{2,3} cannot embed into V.

Abstract

We prove a variety of results about subgroups of Thompson's group $V$. First we prove that every action graph of a finitely generated subgroup of $V$ acting on an orbit in Cantor space is quasi-isometric to a tree. Then we prove that for a broad class of groups of homeomorphisms of the real line, for example Thompson's group $F$, any action on the Cantor space via an embedding into Thompson's group $V$ must be semiconjugate to the standard action on the line. Finally, we use this to establish that many such groups cannot embed into $V$; in particular the Stein group $F_{2,3}$ cannot embed in $V$, answering a question of the third author.