Thompson’s Group F

Part of the Compact Textbooks in Mathematics book series (CTM)


R. Thompson’s group F is introduced and explored. Its elements are described as equivalence classes of tree diagrams and also as continuous piecewise linear functions from the unit interval to itself, and the two descriptions are linked. We give a finite presentation of F along with a presentation on infinitely many generators, which leads to a normal form.


  1. 16.
    Brin, M.G., Squier, C.C.: Groups of piecewise linear homeomorphisms of the real line. Invent. Math. 79, 485–498 (1985)MathSciNetCrossRefGoogle Scholar
  2. 18.
    Brown, K.S.: Finiteness properties of groups. J. Pure Appl. Algebra 44, 45–75 (1987). MR0885095 (88m:20110), Zbl 0613.20033MathSciNetCrossRefGoogle Scholar
  3. 20.
    Cannon, J.W., Floyd, W.J.: What is … Thompson’s group? Not. AMS V. 58, 3, 1112–1113 (2011)Google Scholar
  4. 21.
    Cannon, J.W., Floyd, W.J., Parry, W.R.: Introductory notes on Richard Thompson’s groups, L’Enseignement Mathematique. Revue Internationale. IIe Serie 42 3, 215–256 (1996)MathSciNetzbMATHGoogle Scholar
  5. 24.
    Cleary, S., Taback, J.: Combinatorial properties of Thompson’s group F. Trans. Am. Math. Soc. 356(7), 2825–2849 (2004)MathSciNetCrossRefGoogle Scholar
  6. 35.
    Fordham, S.B.: Minimal length elements of Thompson’s group F. Geom. Dedicata 99, 179–220 (2003)MathSciNetCrossRefGoogle Scholar
  7. 45.
    Halverson, J.: On the dead end depth of Thompson’s Group F. Furman Univ. Elec. J. Undergraduate Math. 15, 5–19 (2011)Google Scholar
  8. 46.
    Hartman, Y., Juschenko, K., Tamuz, O., Ferdowsi, P.V.: Thompson’s group F is not strongly amenable. Ergodic Theory Dynam. Syst. 1–5 (2017).
  9. 48.
    Higman, G.: Finitely Presented Infinite Simple Groups. Notes on Pure Mathematics 8, Department of Pure Mathematics, Department of Mathematics, I.A.S. Australian National University, Canberra (1974). MR0376874 (51 #13049)Google Scholar
  10. 49.
    Horak, M., Stein, M., Taback, J.: Computing word length in alternate forms of Thompson’s group F (2008). arXiv:0706.3218v2Google Scholar
  11. 64.
    Moore, J: Nonassociative Ramsey theory and the amenability of Thompson’s group F. (2012). arXiv:1209.2063v4 [math.GR]Google Scholar
  12. 78.
  13. 81.
    Wajnryb, B., Witowicz, P.: Richard Thompson group F is not amenable.
  14. 85.
    Wladis, C.W.: Metric properties of Thompson’s groups F(n) and F(n,m). Ph.D. thesis The Graduate School and University Center, CUNY, (ProQuest, 2007)Google Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Math and Computer ScienceNew York City College of Technology, The City University of New YorkBrooklynUSA
  2. 2.Department of MathematicsBorough of Manhattan Community College, The City University of New YorkNew YorkUSA
  3. 3.Department of MathematicsMonroe Community CollegeRochesterUSA

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