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The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval

In the 1960s, Richard J. Thompson introduced a triple of groups F ⊆ T ⊆ G which, among them, supplied the first examples of infinite, finitely presented, simple groups [14] (see [6] for published details), a technique for constructing an elementary example of a finitely presented group with an unsolvable word problem [12], the universal obstruction to a problem in homotopy theory [8], and the first examples of torsion free groups of type FP∞ and not of type FP [5]. In abstract measure theory, it has been suggested by Geoghegan (see [3] or [9, Question 13]) that F might be a counterexample to the conjecture that any finitely presented group with no non‐cyclic free subgroup is amenable (admits a bounded, non‐trivial, finitely additive measure on all subsets that is invariant under left multiplication). Recently, F has arisen in the theory of groups of diagrams over semigroup presentations [10], and as the object of questions in the algebra of string rewriting systems [7]. For more extensive bibliographies and more results on Thompson's groups and their generalizations see [1, 4, 6].