A geometric classification of some solvable groups of homeomorphisms

We investigate subgroups of the group PL o ( 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 PL o ( I ) and of the F n . We also show that any standard restricted wreath product (of non‐trivial groups) that embeds in PL o ( I ) or F n must have T ≅ ℤ.