A computational approach to the Thompson group F

Abstract

Communicated by M. SapirLet 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 towhere in both cases the norm of an element in the group ring CF is computed in B( 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, thatIt is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of F .

Computational Explorations of the Thompson Group T for the Amenability Problem of F

“…It is a long standing open problem whether the Thompson group F is amenable. The present paper is a continuation of the work started in [10] by the same authors.…”

“…where S := D 2 and C, and D are the generators of T shown in Figure 1. Let τ be the trace on CT coming from the group von Neumann algebra L(T ), as described in [10,Section 2]. Using the trace properties of τ , the even moments for the self-adjoint operator…”

“…where ||h|| = ||h|| ≤ 2 + √ 2 = 1 + q by Theorem 2.4. In this subsection, we will estimate the spectral distribution measure µh, as it was done in [10,Section 5]. The Hilbert space L 2 ([−(q + 1), (q + 1)], dt) can be equipped with the orthonormal basis…”

“…A recent experimental work by Elder, Rechnitzer and Janse van Reusburg [7] on the amenability problem of F using statistical methods arrives also to the same conclusion that F might be non-amenable (see also [8]). As in [10,Section 2], by the norm of ||a|| of an element a in the group ring of a discrete group Γ we mean…”

“…In this paper we show that if A, B, C denote the standard generators of Thompson group T and D := CBA −1 thenMoreover, the upper bound is attained if the Thompson group F is amenable.Here, the norm of an element in the group ring CT is computed in B( 2 (T )) via the regular representation of T . Using the "cyclic reduced" numbers τ (((C + C 2 )(D + D 2 + D 3 )) n ), n ∈ N, and some methods from our previous paper [10] we can obtain precise lower bounds as well as good estimates of the spectral distributions of 1 12 ((I+C+C 2 )(I+D+D 2 +D 3 )) * (I+C+C 2 )(I+D+D 2 +D 3 ), where τ is the tracial state on the group von Neumann algebra L(T ). Our extensive numerical computations suggest that 1 √ 12and thus that F might be non-amenable.…”