  The objective is to prove the asynchronous exponential growth of the
growth-fragmentation equation in large weighted $L^1$ spaces and under general
assumptions on the coefficients. The key argument is the creation of moments
for the solutions to the Cauchy problem, resulting from the unboundedness of
the total fragmentation rate. It allows us to prove the quasi-compactness of
the associated (rescaled) semigroup, which in turn provides the exponential
convergence toward the projector on the Perron eigenfunction.
