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The ubiquity of Thompson’s group $$F$$ in groups of piecewise linear homeomorphisms of the unit interval. (English) Zbl 0957.20025
In 1973, Richard Thompson constructed the first example of the finitely presented infinite simple group $$G$$. One ingredient of this group $$G$$, a torsion-free subgroup $$F\leq G$$, became popular in 1984 when K. S. Brown and Ross Geoghegan proved the most remarkable fact that $$F$$ – although of infinite cohomological dimension – does admit a $$K(F,1)$$-complex with finite skeleta.
Thompson’s groups have served as counterexamples to various short sighted conjectures. More interestingly: they have the persistent pecularity to pop up in diverse areas in mathematics. This suggested to the authors that there might be something very natural about them. They support this by showing that every group of finitary $$PL$$-homeomorphisms of the unit interval satisfying a certain weak condition contains a copy of Thompson’s group $$F$$ – not unexpected but by no means obvious.
For an introduction to Thompson’s groups see J. W. Cannon, W. J. Floyd and W. R. Parry [Enseign. Math., II. Sér. 42, No. 3/4, 215-256 (1996; Zbl 0880.20027)].

##### MSC:
 20F38 Other groups related to topology or analysis 20F60 Ordered groups (group-theoretic aspects) 57M07 Topological methods in group theory 20F05 Generators, relations, and presentations of groups 20E32 Simple groups 06F15 Ordered groups 20E07 Subgroup theorems; subgroup growth
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