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A computational approach to the Thompson group F

Søren Haagerup

Elmebakken 15, 5260 Odense S, Denmark

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Uffe Haagerup

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark

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, and

Maria Ramirez-Solano

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark

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https://doi.org/10.1142/S0218196715500022Cited by:7 (Source: Crossref)

Abstract

Let F denote the Thompson group with standard generators A = x₀, B = x₁. It is a long standing open problem whether F is an amenable group. By a result of Kesten from 1959, amenability of F is equivalent to

and to

where in both cases the norm of an element in the group ring ℂF is computed in B(ℓ²(F)) via the regular representation of F. By extensive numerical computations, we obtain precise lower bounds for the norms in (i) and (ii), as well as good estimates of the spectral distributions of (I+A+B)*(I+A+B) and of A+A^-1+B+B^-1 with respect to the tracial state τ on the group von Neumann Algebra L(F). Our computational results suggest, that

It is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of F.

Keywords:

AMSC: 20F65, 20-04, 43A07, 22D25, 46L05

References

C. A. Akemann and P. A. Ostrand , Amer. J. Math. 98 , 1015 ( 1976 ) . Crossref, Web of Science, Google Scholar
T. M. Apsotol , Introduction to Analytic Number Theory ( Springer-Verlag , 1976 ) . Google Scholar
G. N. Arzhantseva et al. , Annales Math. Blaise Pascal 15 , 233 ( 2008 ) . Crossref, Google Scholar
J. Belk and K. Brown, Internat. J. Algebra Comput. 15(6), 815 (2005). Link, Web of Science, Google Scholar
M. G. Brin and C. C. Squier , Invent. Math. 79 , 485 ( 1985 ) . Crossref, Web of Science, Google Scholar
N. P. Brown and N. Ozawa , C*-Algebras and Finite-Dimensional Approximations , Graduate Studies in Mathematics 88 ( American Mathematical Society , 2008 ) . Crossref, Google Scholar
J. Burillo, S. Cleary and B. Wiest, Geometric Group Theory, Trends in Mathematics (Birkhäuser, Basel, 2007) pp. 21–35. Crossref, Google Scholar
J. W. Cannon, W. J. Floyd and W. R. Parry, Enseign. Math. (2) 42(4), 215 (1996). Web of Science, Google Scholar
P. Cartiér , Harmonic Analysis on Homogeneous Spaces , Proc. Sym. Pure Mathematics 26 ( American Mathematical Society , 1973 ) . Google Scholar
P.-A. Cherix et al. , Groups with the Haagerup Property, Gromov's a-T-Amenability , Progress in Mathematics 197 ( Birkhaüser , 2001 ) . Crossref, Google Scholar
J. M. Cohen , J. Funct. Analysis 48 , 301 ( 1982 ) . Crossref, Web of Science, Google Scholar
M. Elder, E. Fusy and A. Rechnitzer, J. Algebra 324(1), 102 (2010). Crossref, Web of Science, Google Scholar
M. Elder, A. Rechnitzer and E. J. Janse van Reusburg, Random sampling of trivial words in finitely presented groups, preprint (2013) , arXiv:math/1312.5722v2 . Google Scholar
M. Elder, A. Rechnitzer and T. Wong, Groups Complex. Cryptol. 4(2), 301 (2012). Crossref, Google Scholar
D. S. Farley , Int. Math. Res. Notices 45 , 2409 ( 2003 ) . Crossref, Web of Science, Google Scholar
R. I. Grigorchuk, Multicomponent Random Systems, eds. R. L. Dobrushin and Ya. G. Sinai (Marcel Dekker, New York-Basel, 1980) pp. 285–325. Google Scholar
V. S. Guba, Internat. J. Algebra Comput. 14(6), 677 (2004). Link, Web of Science, Google Scholar
U. Haagerup and S. Knudby , Proc. Amer. Math. Soc. ( 2014 ) , DOI: 10.1090/S0002-9939-2014-12466-X . Google Scholar
U. Haagerup and K. K. Olesen, The Thompson group T and V are not inner amenable, in preparation (2014) . Google Scholar
A. Erdélyi (ed.) , Higher Transcendental Functions II ( Mc. Graw-Hill , 1953 ) . Google Scholar
N. Higson and G. Kasparov , Invent. Math. 144 , 23 ( 2001 ) . Crossref, Web of Science, Google Scholar
R. V. Kadison and J. Ringrose , Fundamentals of the Theory of Operator Algebras 2 ( American Mathematical Society , 1986 ) . Google Scholar
H. Kesten , Math. Scand. 7 , 146 ( 1959 ) . Crossref, Google Scholar
H. Kesten , Trans. Amer. Math. Soc. 92 , 336 ( 1959 ) . Crossref, Google Scholar
F. Lehner , Proc. Amer. Math. Soc. 125 , 3423 ( 1997 ) . Crossref, Web of Science, Google Scholar
M. Leinert , Studia Math. 52 , 149 ( 1974 ) . Crossref, Web of Science, Google Scholar
N. Monod, Proc. Natl. Acad. Sci. USA 110(12), 4524 (2013). Crossref, Web of Science, Google Scholar
G. J. Murphy , C*-Algebras and Operator Theory ( Academic Press , 1990 ) . Google Scholar
G. K. Pedersen , Analysis Now , Graduate Texts in Mathematics 118 ( Springer , 1988 ) . Google Scholar
M. Puschnigg , Invent. Math. 149 , 153 ( 2002 ) . Crossref, Web of Science, Google Scholar
M. Reed and B. Simon , Methods of Modern Mathematical Physics I: Functional Analysis ( Academic Press , 1980 ) . Google Scholar
S. Sawyer , Zeithschirft Wahrscheirnlécheitsth 42 , 279 ( 1978 ) . Crossref, Google Scholar
G. Szego , Orthonormal Polynomials , 3rd edn. , Colloq. Publ 23 ( Amer. Math. Soc. , 1967 ) . Google Scholar