  We study growth rates for strongly continuous semigroups. We prove that a
growth rate for the resolvent on imaginary lines implies a corresponding growth
rate for the semigroup if either the underlying space is a Hilbert space, or
the semigroup is asymptotically analytic, or if the semigroup is positive and
the underlying space is an $L^{p}$-space or a space of continuous functions. We
also prove variations of the main results on fractional domains; these are
valid on more general Banach spaces. In the second part of the article we apply
our main theorem to prove optimality in a classical example by Renardy of a
perturbed wave equation which exhibits unusual spectral behavior.
