Amenability problem for Thompson's group \(F\): state of the art
Thompson's group \(F\) is one of the three famous groups defined and studied by R. Thompson in some unpublished notes in the 1960s.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 \(K\).
scientific article; zbMATH DE number 7910106
scientific article