© 1999 by London Mathematical Society
© The London Mathematical Society
The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval
Department of Mathematical Sciences, State University of New York at Binghamton Binghamton, NY 13902-6000, USA matt{at}math.binghamton.edu
Received 22 April 1997. Revision received 19 December 1997.
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].
A persistent peculiarity of Thompson's groups is their ability to pop up in diverse areas of mathematics. This suggests that there might be something very natural about Thompson's groups. We support this idea by showing (Theorem 1.1 below) that PLo(I), the group of piecewise linear (finitely many changes of slope), orientation-preserving, self-homeomorphisms of the unit interval, is riddled with copies of F: a very weak criterion implies that a subgroup of PLo(I) must contain an isomorphic copy of F.