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

Abstract

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 $D:=CB{A}^{-1}$ then $$\sqrt{2}+\sqrt{3} < \frac{1}{\sqrt{12}}\left|\right|(I+C+C^2)(I+D+D^2+D^3)\left|\right| \le 2+\sqrt{2}.$$Moreover, the upper bound is attained if the Thompson group F is amenable. Here, the norm of an element in the group ring $\mathbb{C}T$ is computed in $B({\ell }^2(T))$ via the regular representation of T. Using the “cyclic reduced” numbers $\tau ({((C+C^2)(D+D^2+D^3))}^n),n\in \mathbb{N}$, 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 $\frac{1}{12}{((I+C+C^2)(I+D+D^2+D^3))}^{*}(I+C+C^2)(I+D+D^2+D^3),$ where τ is the tracial state on the group von Neumann algebra L(T). Our extensive numerical computations suggest that $$\frac{1}{\sqrt{12}}\left|\right|(I+C+C^2)(I+D+D^2+D^3)\left|\right|\approx 3.28,$$and, thus that F might be non-amenable. However, we can in no way rule out that $\frac{1}{\sqrt{12}}\left|\right|(I+C+C^2)(I+D+D^2+D^3)\left|\right|= 2+\sqrt{2}$.

Keywords:

• Thompson’s groups F, T
• estimating norms of products in group C^*-algebras
• amenability
• cogrowth
• computer calculations

Acknowledgments

We are very grateful with the DeIC National HPC Centre for allowing us to use the supercomputer Abacus 2.0 to run some of our computer code. The first and third author would like to thank Steen Thorbjørnsen, Erik Christensen, and Wojciech Szymanski for the nice discussions that they had with them. The second author was supported by the Villum Foundation under the project “Local and global structures of groups and their algebras” at University of Southern Denmark, and by the ERC Advanced Grant no. OAFPG 247321, and partially supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at University of Copenhagen, and the Danish Council for Independent Research, Natural Sciences. The third author was supported by the Villum Foundation under the project “Local and global structures of groups and their algebras” at University of Southern Denmark, and by the ERC Advanced Grant no. OAFPG 247321, and by the Center for Experimental Mathematics at University of Copenhagen.

Additional information

Funding

Supported by the Villum Foundation under the project “Local and global structures of groups and their algebras” at University of Southern Denmark, and by the ERC Advanced Grant no. OAFPG 247321, and partially supported by the Danish National Research Foundation (DNRF) through the Centre for Symmetry and Deformation at University of Copenhagen, and the Danish Council for Independent Research, Natural Sciences.