Infinite periodic words and almost nilpotent varieties

An almost nilpotent variety of linear growth is constructed in the paper for any infinite periodic word in an alphabet of two letters. A discrete series of different almost nilpotent varieties is also constructed. Only a few almost nilpotent varieties were studied previously and their existence was proved often under some additional assumptions. It was proved the existence of almost nilpotent varieties of any integer exponent with a fractional exponent, as well as the existence of a continual family of almost nilpotent varieties with not more than quadratic growth.