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We construct finitely generated nil algebras with prescribed growth rate. Inparticular, any increasing submultiplicative function is realized as the growthfunction of a nil algebra up to a polynomial error term and an arbitrarily slowdistortion. We then move on to examples of nil algebras and domains withstrongly oscillating growth functions and construct primitive algebras forwhich the Gelfand-Kirillov dimension is strictly sub-additive with respect totensor products, thus answering a question raised by Krempa-Okninski andKrause-Lenagan.

Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth