Amenability problem for Thompson's group \(F\): state of the art
type: Item
instance of: scholarly article
title: Amenability problem for Thompson's group \(F\): state of the art
author: Victor S. Guba
published in: Journal of Groups, Complexity, Cryptology
publication date: 2024-09-10T00:00:00Z
review text: Thompson's group \(F\) is one of the three famous groups defined and studied by R. Thompson in some unpublished notes in the 1960s. The group \(F\), is the group of all piecewise-linear homeomorphisms of the interval \([0,1]\) with finitely many breakpoints, where all breakpoints are finite dyadic fractions and all slopes are integer powers of 2. It is finitely presented: \(F=\big \langle x_{0}, x_{1} \; \big | x_{1}^{x_{0}^{2}}=x_{1}^{x_{0}x_{1}}, x_{1}^{x_{0}^{3}}=x_{1}^{x_{0}^{2}x_{1}} \; \big \rangle\). The problem of amenability for \(F\) was posed by \textit{R. Geoghegan} in [Topol. Proc. 4, No. 1, 287--338 (1980; Zbl 0448.57001)].\N\NThe paper under review is a survey of author's recent results (published over the two decades 2004--2024) on the amenability problem for \(F\). They in particular concern esimating the density of finite subgraphs in Cayley graphs of \(F\) for various systems of generators, and also equations in the group ring \(K[F]\) of \(F\) over a field \(K\).
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 16U20
zbMATH DE Number: 7910106
zbMATH Keywords: amenable group
zbMATH Keywords: Thompson's group \(F\)
zbMATH Keywords: Cayley graph
zbMATH Keywords: group ring
MaRDI profile type: Publication
cites work: Q3657123
cites work: Testing Cayley graph densities.
cites work: Groups of piecewise linear homeomorphisms of the real line
cites work: Finiteness properties of groups
cites work: An infinite-dimensional torsion-free \(\text{FP}_{\infty}\) group
cites work: Introductory notes on Richard Thompson's groups
cites work: Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces
cites work: Elementary amenable groups
cites work: Amenable semigroups
cites work: Automaticity, goals, and environmental interactions.
cites work: Ruinous subsets of Richard Thompson's group \(F\).
cites work: A partial converse to a theorem of Tamari.
cites work: A Minimal Candidate for the Existence of a Weakly-Ruinous Set
cites work: On groups with full Banach mean value
cites work: Q3894661
cites work: Q4724819
cites work: Q5562332
cites work: Q3930365
cites work: Q3350136
cites work: An example of a finitely presented amenable group not belonging to the classEG
cites work: Diagram groups
cites work: ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F
cites work: On the density of Cayley graphs of R.Thompson’s group F in symmetric generators
cites work: Amenability of semigroups and the Ore condition for semigroup rings
cites work: On the Ore condition for the group ring of R. Thompson’s group F
cites work: R. Thompson’s group and the amenability problem
cites work: Systems of equations over the group ring of Thompson’s group F
cites work: Evacuation schemes on Cayley graphs and non-amenability of groups
cites work: Symmetric Random Walks on Groups
cites work: Full Banach Mean Values on Countable groups.
cites work: Über eine Schlußweise aus dem Endlichen ins Unendliche.
cites work: Fast growth in the Følner function for Thompson's group \(F\).
cites work: Zur allgemeinen Theorie des Maßes.
cites work: ON THE PROBLEM OF THE EXISTENCE OF AN INVARIANT MEAN ON A GROUP
cites work: Non-amenable finitely presented torsion-by-cyclic groups.
cites work: Combinatorial Algebra: Syntax and Semantics
cites work: Q3341041
DOI: 10.46298/JGCC.2023.15.1.11315
description: scientific article; zbMATH DE number 7910106