Solvability in Groups of Piecewise-linear Homeomorphisms of the Unit   Interval
  We investigate subgroups of the group PLo(I) of piecewise-linear, orientation preserving homeomorphisms of the unit interval with finitely many breaks in slope, and also subgroups of Thompson's group F. We find geometric criteria determining the derived length of any such group, and use this criteria to classify the solvable and non-solvable subgroups of PLo(I) and of F.   Let H be a subgroup of PL_o(I) or F. We find that H is solvable if and only if H is isomorphic to a group in a well described class R of groups. We also find that H is non-solvable if and only if we can embed a copy of a specific non-solvable group W into H.   We strengthen the non-solvability classification by finding weak geometric criteria under which we can embed other groups (all containing W) into non-solvable subgroups of PLo(I) or F. 
Collin Bleak