If A is a graded connected algebra then we define a new invariant, polydepth
A, which is finite if ExtAââ(M,A)î =0 for some A-module M of at most
polynomial growth. Theorem 1: If f : X \to Y is a continuous map of finite
category, and if the orbits of H_*(\Omega Y) acting in the homology of the
homotopy fibre grow at most polynomially, then H_*(\Omega Y) has finite
polydepth. Theorem 2: 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, âi=k+1k+dâdimLiââ„kr, kâ„ some
k(r)