TY - JOUR

T1 - Computational Explorations of the Thompson Group T for the Amenability Problem of F

AU - Haagerup, Søren

AU - Haagerup, Uffe

AU - Ramirez-Solano, Maria

PY - 2019/8/30

Y1 - 2019/8/30

N2 - It is a long standing open problem whether the Thompson group F is an amenable group. In this article, we show that if A, B, C denote the standard generators of Thompson group T and (Formula presented.) then (Formula presented.) Moreover, the upper bound is attained if the Thompson group F is amenable. Here, the norm of an element in the group ring (Formula presented.) is computed in (Formula presented.) via the regular representation of T. Using the “cyclic reduced” numbers (Formula presented.), and some methods from our previous article [Haagerup et al. 15] we can obtain precise lower bounds as well as good estimates of the spectral distributions of (Formula presented.) where τ is the tracial state on the group von Neumann algebra L(T). Our extensive numerical computations suggest that (Formula presented.) and, thus that F might be non-amenable. However, we can in no way rule out that (Formula presented.).

AB - It is a long standing open problem whether the Thompson group F is an amenable group. In this article, we show that if A, B, C denote the standard generators of Thompson group T and (Formula presented.) then (Formula presented.) Moreover, the upper bound is attained if the Thompson group F is amenable. Here, the norm of an element in the group ring (Formula presented.) is computed in (Formula presented.) via the regular representation of T. Using the “cyclic reduced” numbers (Formula presented.), and some methods from our previous article [Haagerup et al. 15] we can obtain precise lower bounds as well as good estimates of the spectral distributions of (Formula presented.) where τ is the tracial state on the group von Neumann algebra L(T). Our extensive numerical computations suggest that (Formula presented.) and, thus that F might be non-amenable. However, we can in no way rule out that (Formula presented.).

KW - amenability

KW - cogrowth

KW - computer calculations

KW - estimating norms of products in group C-algebras

KW - Thompson’s groups F, T

U2 - 10.1080/10586458.2018.1502699

DO - 10.1080/10586458.2018.1502699

M3 - Journal article

AN - SCOPUS:85071391999

JO - Experimental Mathematics

JF - Experimental Mathematics

SN - 1058-6458

ER -