A Geometric Classification of Some Solvable Groups of Homeomorphisms

We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of generalized Thompson groups Fn. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the solvable and non-solvable subgroups of PLo(I) and of the Fn. We also show that any standard restricted wreath product C ≀T (of non-trivial groups) that embeds in PLo(I) or Fn must have T ∼= Z.