A computational approach to the Thompson group F release_2r6h526xxrcn5efubfoaay47pi

by S. Haagerup, U. Haagerup, M. Ramirez-Solano

Released as a report .

(2014)

Abstract

Let F denote the Thompson group with standard generators A=x_0, B=x_1. 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 (i) ||I+A+B||=3 and to (ii) ||A+A^-1+B+B^-1||=4, where in both cases the norm of an element in the group ring C F is computed in B(ℓ^2(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 ||I+A+B||≈ 2.95 ||A+A^-1+B+B^-1||≈ 3.87. It is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of F.
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Type  report
Stage   submitted
Date   2014-09-04
Version   v1
Language   en ?
Number  CPH-SYM-DNRF92
arXiv  1409.1486v1
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