  In previous work, the authors established various bounds for the dimensions
of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of
semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig)
quantum groups $U_\zeta$ (at roots of unity $\zeta$). These bounds depend only
on the root system, and not on the characteristic $p$ or the size of the root
of unity $\zeta$. This paper investigates the rate of growth of these bounds.
Both in the quantum and algebraic group situation, these rates of growth
represent new and fundamental invariants attached to the root system $\Phi$.
For quantum groups $U_\zeta$ with a fixed $\Phi$, we show the sequence
  $\{\max_{L\,{\text{irred}}}\dim \opH^n(U_\zeta,L)\}_n$ has polynomial growth
independent of $\zeta$. In fact, we provide upper and lower bounds for the
polynomial growth rate. Applications of these and related results for
$\Ext^n_{U_\zeta}$ are given to Kazhdan-Lusztig polynomials. Polynomial growth
in the algebraic group case remains an open question, though it is proved that
$\{\log\max_{L\,\text{\rm irred}}\dim\opH^n(G,L)\}$ has polynomial growth $\leq
3$ for any fixed prime $p$ (and $\leq 4$ if $p$ is allowed to vary with $n$).
  We indicate the relevance of these issues to (additional structure for) the
constants proposed in the theory of higher cohomology groups for finite simple
groups with irreducible coefficients by Guralnick, Kantor, Kassabov, and
Lubotzky \cite{GKKL}.
