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 F_n. We find geometric criteria determining the derived length of any such group, and use these criteria to produce a geometric classification of the solvable and non-solvable subgroups of PLo(I) and of the F_n. We also show that any standard restricted wreath product (of non-trivial groups) that embeds in PLo(I) or F_n must have T≅ℤ.