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

Søren Haagerup, Uffe Haagerup, Maria Ramirez-Solano*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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 (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.).

Original languageEnglish
JournalExperimental Mathematics
Volume30
Issue number1
Pages (from-to)105-126
ISSN1058-6458
DOIs
Publication statusPublished - 2021

Keywords

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

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