September 10, 2003
Thompson's group F, quasi-isometric embeddings
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 $\phi$ 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 $F^m \times \Z^n$, for integral $m,n \geq 0$,
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.
Both authors thank the referee for helpful comments.
Sean Cleary :
Department of Mathematics, City College of New York, City University of New York, New York, NY 10031
Department of Mathematics and Statistics, University at Albany, Albany, NY 12222