Abstract : 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.