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 In unpublished notes, Thompson proved that \(T\) and \(V\) are finitely-presented, infinite simple groups.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 They were expanded in order to make available Thompson’s unpublished proof of the simplicity of \(T\) and \(V\) and Thurston’s interpretations of \(F\) and \(T\) as the groups of orientation-preserving, piecewise integral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

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 They were expanded in order to make available Thompson’s unpublished proof of the simplicity of \(T\) and \(V\) and Thurston’s interpretations of \(F\) and \(T\) as the groups of orientation-preserving, piecewise integral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.

Reviewer: L.A.Bokut’ (Novosibirsk)

### 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 |

### Keywords:

amenable groups; finitely-presented groups; unsolvable word problems; finitely-presented, infinite simple groups; finitely generated simple subgroups; orientation-preserving, piecewise integral projective homeomorphisms
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\textit{J. W. Cannon} et al., Enseign. Math. (2) 42, No. 3--4, 215--256 (1996; Zbl 0880.20027)

### Online Encyclopedia of Integer Sequences:

Self-inverse permutation of natural numbers induced by Catalan Automorphism *A072796 acting on the parenthesizations encoded by A014486.Signature permutation of a Catalan bijection: row 3655 of A089840.

Signature permutation of a Catalan bijection: row 3656 of A089840.

Self-inverse signature permutation of a Catalan bijection: row 183 of A089840.