  We show that Golod-Shafarevich algebras can be homomorphically mapped onto
infinite-dimensional algebras with polynomial growth, under mild assumptions of
the number of relations of given degrees.
  In case these algebras are finitely presented, we show they can be mapped
onto an infinite dimensional algebras with quadratic growth. This answers a
guestion by Zelmanov.
  We then show, by an elementary construction, that any sufficiently regular
function at least $n^{\log n}$ may occur as the growth of an algebra.
