Metrics and embeddings of generalizations of Thompson’s group $F$

Abstract

The distance from the origin in the word metric for generalizations $F(p)$ of Thompson’s group $F$ is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of $F(p)$. This interpretation of the metric is used to prove that every $F(p)$ admits a quasi-isometric embedding into every $F(q)$, and also to study the behavior of the shift maps under these embeddings.