  We are interested in the cycles obtained by slicing at all heights random
Boltzmann triangulations with a simple boundary. We establish a functional
invariance principle for the lengths of these cycles, appropriately rescaled,
as the size of the boundary grows. The limiting process is described using a
self-similar growth-fragmentation process with explicit parameters. To this
end, we introduce a branching peeling exploration of Boltzmann triangulations,
which allows us to identify a crucial martingale involving the perimeters of
cycles at given heights. We also use a recent result concerning self-similar
scaling limits of Markov chains on the nonnegative integers. A motivation for
this work is to give a new construction of the Brownian map from a
growth-fragmentation process.
