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 F^m×Zⁿ, for integral m,n ≧ 0, which were shown to occur as quasi-isometrically embedded subgroups by Burillo and Guba and Sapir.

The first author acknowledges support from PSC-CUNY grant #63438-0032.

The second author acknowledges partial support from an NSF-AWM Mentoring Travel Grant and would like to thank the University of Utah for its hospitality during the writing of this paper.