A computational approach to the Thompson group F

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 CF 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 tau on the group von Neumann Algebra L(F). Our computational results suggest, that \\I + A + B\\ approximate to 2.95 \\A + A(-1) + B + B-1\\ approximate to 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.