  Given a Baumslag-Solitar group, we study its space of subgroups from a
topological and dynamical perspective. We first determine its perfect kernel
(the largest closed subset without isolated points). We then bring to light a
natural partition of the space of subgroups into one closed subset and
countably many open subsets that are invariant under the action by conjugation.
One of our main results is that the restriction of the action to each piece is
topologically transitive. This partition is described by an arithmetically
defined function, that we call the phenotype, with values in the positive
integers or infinity. We eventually study the closure of each open piece and
also the closure of their union. We moreover identify in each phenotype a (the)
maximal compact invariant subspace.
