Let a>=2 be a real number and k = [a]. We denote by Diff^a(I) the group of C^k diffeomorphisms of a closed interval I such that the k--th derivatives are H将モlder--continuous of exponent (a - k). For each real number a>=2, we construct a finitely generated group G < Diff^a(I) such that G admits no injective homomorphisms into Diff^b(I) for any b>a. We also construct a countable simple group with the same property. This is a joint work with Thomas Koberda.