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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 In unpublished notes, Thompson proved that $$T$$ and $$V$$ are finitely-presented, infinite simple groups.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 They were expanded in order to make available Thompson’s unpublished proof of the simplicity of $$T$$ and $$V$$ and Thurston’s interpretations of $$F$$ and $$T$$ as the groups of orientation-preserving, piecewise integral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.

##### MSC:
 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