On some generalizations of abelian power avoidability

We prove that 2-abelian-cubes are avoidable over a binary alphabet and that 3-abelian-squares are avoidable over a ternary alphabet, answering positively to two questions of Karhumäki et al.. We also show the existence of infinite additive-cube-free words on several ternary alphabets. To achieve this, we give sufficient conditions for a morphism to be k-abelian-n-power-free (resp. additive-n-power-free), and then we give several morphisms which respect these conditions. Additionally, all our constructions show that the number of such words grows exponentially. As a corollary, we get a new lower bound of 3 = 1.059526 . . . for the growth rate of abelian-cube-free words.