Introductory notes on Richard Thompson’s groups. (English) Zbl 0880.20027

The groups \(F\), \(T\) and \(V\) were defined by Richard Thompson in 1965. McKenzie and Thompson used them in 1977 to construct finitely-presented groups with unsolvable word problems. In unpublished notes, Thompson proved that \(T\) and \(V\) are finitely-presented, infinite simple groups. Thompson used \(V\) in his proof that a finitely-presented group has solvable word problem if and only if it can be embedded into a finitely generated simple subgroup of a finitely presented group. The group \(F\) was rediscovered and studied by homotopy theorists. Higman (1974) generalized \(V\) to an infinite family of finitely presented simple groups.
These notes originate from the authors’ interest in the question of whether or not \(F\) is amenable. They were expanded in order to make available Thompson’s unpublished proof of the simplicity of \(T\) and \(V\) and Thurston’s interpretations of \(F\) and \(T\) as the groups of orientation-preserving, piecewise integral projective homeomorphisms of the unit interval and the circle.


20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E32 Simple groups
20F05 Generators, relations, and presentations of groups
20F24 FC-groups and their generalizations
43A07 Means on groups, semigroups, etc.; amenable groups