  In a growth-fragmentation system, cells grow in size slowly and split apart
at random. Typically, the number of cells in the system grows exponentially and
the distribution of the sizes of cells settles into an equilibrium 'asymptotic
profile'. In this work we introduce a new method to prove this asymptotic
behaviour for the growth-fragmentation equation, and show that the convergence
to the asymptotic profile occurs at exponential rate. We do this by identifying
an associated sub-Markov process and studying its quasi-stationary behaviour
via a Lyapunov function condition. By doing so, we are able to simplify and
generalise results in a number of common cases and offer a unified framework
for their study. In the course of this work we are also able to prove the
existence and uniqueness of solutions to the growth-fragmentation equation in a
wide range of situations.
