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Some Graphs Related to Thompson’s Group F

  • Dmytro Savchuk
Part of the Trends in Mathematics book series (TM)

Abstract

The Schreier graphs of Thompson’s group F with respect to the stabilizer of 1/2 and generators x 0 and x 1 , and of its unitary representation in L 2 ([0, 1]) induced by the standard action on the interval [0, 1] are explicitly described. The coamenability of the stabilizers of any finite set of dyadic rational numbers is established. The induced subgraph of the right Cayley graph of the positive monoid of F containing all the vertices of the form x n v, where n ≥ 0 and v is any word over the alphabet {x 0 , x 1 }, is constructed. It is proved that the latter graph is non-amenable.

Mathematics Subject Classification (2000)

20F65 

Keywords

Thompson’s group amenability Schreier graphs Cayley graphs 

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References

  1. [BdlHV08]
    Bachir Bekka, Pierre de la Harpe, and Alain Valette. Kazhdan’s property (T), volume 11 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008.Google Scholar
  2. [Bel04]
    James M. Belk. Thompson’s group F. PhD thesis, Cornell University, 2004.Google Scholar
  3. [Bro87]
    Kenneth S. Brown. Finiteness properties of groups. In Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), volume 44, pages 45–75, 1987.zbMATHGoogle Scholar
  4. [BS85]
    Matthew G. Brin and Craig C. Squier. Groups of piecewise linear homeomorphisms of the real line. Invent. Math., 79(3):485–498, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [Bur99]
    José Burillo. Quasi-isometrically embedded subgroups of Thompson’s group F. J. Algebra, 212(1):65–78, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [CFP96]
    J.W. Cannon, W.J. Floyd, and W.R. Parry. Introductory notes on Richard Thompson’s groups. Enseign. Math. (2), 42(3–4):215–256, 1996.zbMATHMathSciNetGoogle Scholar
  7. [Dau92]
    Ingrid Daubechies. Ten lectures on wavelets, volume 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992.Google Scholar
  8. [dlAGCS99]
    P. de lya Arp, R.I. Grigorchuk, and T. Chekerini-Silberstaįn. Amenability and paradoxical decompositions for pseudogroups and discrete metric spaces. Tr. Mat. Inst. Steklova, 224 (Algebra. Topol. Differ. Uravn. i ikh Prilozh.):68–111, 1999.Google Scholar
  9. [Don07]
    John Donnelly. Ruinous subsets of Richard Thompson’s group F. J. Pure Appl. Algebra, 208(2):733–737, 2007.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [Gri90]
    R.I. Grigorchuk. Growth and amenability of a semigroup and its group of quotients. In Proceedings of the International Symposium on the Semigroup Theory and its Related Fields (Kyoto, 1990), pages 103–108, Matsue, 1990 Shimane Univ.Google Scholar
  11. [Gri98]
    R.I. Grigorchuk. An example of a finitely presented amenable group that does not belong to the class EG. Mat. Sb., 189(1):79–100, 1998.zbMATHMathSciNetGoogle Scholar
  12. [GS87]
    S.M. Gersten and John R. Stallings, editors. Combinatorial group theory and topology, volume 111 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1987. Papers from the conference held in Alta, Utah, July 15–18, 1984.Google Scholar
  13. [GS98]
    R.I. Grigorchuk and A.M. Stepin. On the amenability of cancellation semi-groups. Vestnik Moskov. Univ. Ser. I Mat. Mekh., (3):12–16, 73, 1998.MathSciNetGoogle Scholar
  14. [Haa10]
    Alfred Haar. Zur Theorie der orthogonalen Funktionensysteme. Math. Ann., 69(3):331–371, 1910.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [OS02]
    Alexander Yu. Olshanskii and Mark V. Sapir. Non-amenable finitely presented torsion-by-cyclic groups. Publ. Math. Inst. Hautes Études Sci., (96):43–169 (2003), 2002.Google Scholar
  16. [Šun07]
    Zoran ĽSunić. Tamari lattices, forests and Thompson monoids. European J. Combin., 28(4):1216–1238, 2007.CrossRefMathSciNetGoogle Scholar
  17. [WS01]
    Gilbert G. Walter and Xiaoping Shen. Wavelets and other orthogonal systems. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2001.Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Dmytro Savchuk
    • 1
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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