  A self-similar growth-fragmentation describes the evolution of particles that
grow and split as time passes. Its genealogy yields a self-similar continuum
tree endowed with an intrinsic measure. Extending results of Haas for pure
fragmentations, we relate the existence of an absolutely continuous profile to
a simple condition in terms of the index of self-similarity and the so-called
cumulant of the growth-fragmentation. When absolutely continuous, we
approximate the profile by a function of the small fragments, and compute the
Hausdorff dimension in the singular case. Applications to Boltzmann random
planar maps are emphasized, exploiting recently established connections between
growth-fragmentations and random maps by Bertoin et al.
