A minimal non-solvable group of homeomorphisms

Let PLo(I) represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group W and show that W embeds in every non-solvable subgroup of PLo(I). We find mild conditions under which other non-solvable subgroups B, (≀ℤ≀)∞, (ℤ≀)∞, and ∞(≀ℤ) embed in subgroups of Let PLo(I) represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group W and show that W embeds in every non-solvable subgroup of PLo(I). We show that all solvable subgroups of PLo(I) embed in all non-solvable subgroups of PLo(I). These results continue to apply if we replace PLo(I) by any generalized Thompson group Fn.