IMAGES OF GOLOD-SHAFAREVICH ALGEBRAS WITH SMALL GROWTH

We show that Golod-Shafarevich algebras can be homomorphically mapped onto infinite dimensional algebras with polynomial growth when mild assumptions about the number of relations of given degrees are introduced. This answers a question by Zel'manov. In the case where these algebras are finitely presented, we show that they can be mapped onto infinite-dimensional algebras with at most quadratic growth. We then use an elementary construction to show that any sufficiently regular function a parts per thousand(3) n(log n) may occur as the growth function of an algebra.