  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 the
generalized Thompson groups F_n. 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
F_n. We also show that any standard restricted wreath product C wr T (of
non-trivial groups) that embeds in PLo(I) or F_n must have T isomorphic with
the integers.
