Graded lie algebras with finite polydepth

If A is a graded connected algebra then we define a new invariant, polydepth A, which is finite if Ext*(A) (M, A) not equal 0 for some A-module M of at most polynomial growth. THEOREM 1: If f : X --> Y is a continuous map of finite category, and if the orbits of H*(OmegaY) acting in the homology of the homotopy fibre grow at most polynomially, then H*(OmegaY) has finite polydepth. THEOREM 5: If L is a graded Lie algebra and polydepth UL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, Sigma(i=k+1)(k+d) dim L-i greater than or equal to V, k greater than or equal to some k(r). (C) 2003 Elsevier SAS.