Solvability in Groups of Piecewise-linear Homeomorphisms of the Unit Interval by Collin Bleak

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 PLo(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 .