Volume 77, Nº 2 (2022)

This paper focuses on Richard Thompson's group $F$, which was discovered in the 1960s. Many papers have been devoted to this group. We are interested primarily in the famous problem of amenability of this group, which was posed by Geoghegan in 1979. Numerous attempts have been made to solve this problem in one way or the other, but it remains open.In this survey we describe the most important known properties of this group related to the word problem and representations of elements of the group by piecewise linear functions as well as by diagrams and other geometric objects. We describe the classical results of Brin and Squier concerning free subgroups and laws. We include a description of more modern important results relating to the properties of the Cayley graphs (the Belk–Brown construction) as well as Bartholdi's theorem about the properties of equations in group rings. We consider separately the criteria for (non-)amenability of groups that are useful in the work on the main problem. At the end we describe a number of our own results about the structure of the Cayley graphs and a new algorithm for solving the word problem.Bibliography: 69 titles.