Metrics and embeddings of generalizations of Thompson’s group 𝐹

Source abstract

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