Combinatorial properties of Thompson’s group $F$
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- by Sean Cleary and Jennifer Taback PDF
- Trans. Amer. Math. Soc. 356 (2004), 2825-2849 Request permission
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.References
- Kenneth S. Brown and Ross Geoghegan, An infinite-dimensional torsion-free $\textrm {FP}_{\infty }$ group, Invent. Math. 77 (1984), no. 2, 367–381. MR 752825, DOI 10.1007/BF01388451
- J. Burillo, S. Cleary, and M. I. Stein, Metrics and embeddings of generalizations of Thompson’s group $F$, Trans. Amer. Math. Soc. 353 (2001), no. 4, 1677–1689. MR 1806724, DOI 10.1090/S0002-9947-00-02650-7
- José Burillo, Quasi-isometrically embedded subgroups of Thompson’s group $F$, J. Algebra 212 (1999), no. 1, 65–78. MR 1670622, DOI 10.1006/jabr.1998.7618
- J. W. Cannon, W. J. Floyd, and W. R. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215–256. MR 1426438
- Sean Cleary and Jennifer Taback. Thompson’s group ${F}$ is not almost convex. J. Algebra. to appear.
- Blake Fordham. Minimal Length Elements of Thompson’s group ${F}$. Ph.D. thesis, Brigham Young Univ, 1995.
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
- 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.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 2825-2849
- MSC (2000): Primary 20F65
- DOI: https://doi.org/10.1090/S0002-9947-03-03375-0
- MathSciNet review: 2052598