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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Combinatorial properties of Thompson's group $F$


Authors: Sean Cleary and Jennifer Taback
Journal: Trans. Amer. Math. Soc. 356 (2004), 2825-2849
MSC (2000): Primary 20F65
Published electronically: October 28, 2003
MathSciNet review: 2052598
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Abstract: We study some combinatorial consequences of Blake Fordham's theorems on the word metric of Thompson's group $F$ in the standard two generator presentation. We explore connections between the tree pair diagram representing an element $w$ of $F$, its normal form in the infinite presentation, its word length, and minimal length representatives of it. We estimate word length in terms of the number and type of carets in the tree pair diagram and show sharpness of those estimates. In addition we explore some properties of the Cayley graph of $F$ with respect to the two generator finite presentation. Namely, we exhibit the form of ``dead end'' elements in this Cayley graph, and show that it has no ``deep pockets''. Finally, we discuss a simple method for constructing minimal length representatives for strictly positive or negative words.


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Additional Information

Sean Cleary
Affiliation: Department of Mathematics, City College of New York, City University of New York, New York, New York 10031
Email: cleary@sci.ccny.cuny.edu

Jennifer Taback
Affiliation: Department of Mathematics and Statistics, University at Albany, Albany, New York 12222
Email: jtaback@math.albany.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-03-03375-0
PII: S 0002-9947(03)03375-0
Received by editor(s): August 22, 2002
Received by editor(s) in revised form: March 20, 2003
Published electronically: October 28, 2003
Additional Notes: The first author acknowledges support from PSC-CUNY grant #63438-0032. The second author thanks the University of Utah for their hospitality during the writing of this paper.
Article copyright: © Copyright 2003 American Mathematical Society