Geometric quasi-isometric embeddings into Thompson's group F

We use geometric techniques to investigate several examples of quasi-isometrically embedded subgroups of Thompson’s group F . Many of these are explored using the metric properties of the shift map φ in F . These subgroups have simple geometric but complicated algebraic descriptions. We present them to illustrate the intricate geometry of Thompson’s group F as well as the interplay between its standard finite and infinite presentations. These subgroups include those of the form Fm × Zn, for integral m,n ≥ 0, which were shown to occur as quasi-isometrically embedded subgroups by Burillo and Guba and Sapir.