Groups with ℓm-abelianity

The finitely generated group G is said to be n-abelian if Gn=1, for some positive integer n, where Gn =< [x, yn]|x, y ∈ G >. The groups G = (ℓ, m|n, k) = ⟨a, b | aℓ = bm = (ab)n = (ab−1)k = 1⟩, have been studied completely by M. Edjvet and R. M. Thomas in 1997. In this paper we show that there exist the certain classes with ℓm-abelianity for every positive integer ℓm. Our groups are the (ℓ, m|n, k)-groups and extensions of deficiency zero and −1