  Furstenberg has associated to every topological group $G$ a universal
boundary $\partial(G)$. If we consider in addition a subgroup $H<G$, the
relative notion of $(G,H)$-boundaries admits again a maximal object
$\partial(G,H)$. In the case of discrete groups, an equivalent notion was
introduced by Bearden--Kalantar as a very special instance of their
constructions. However, the analogous universality does not always hold, even
for discrete groups. On the other hand, it does hold in the affine
reformulation in terms of convex compact sets, which admits a universal simplex
$\Delta(G,H)$, namely the simplex of measures on $\partial(G,H)$. We determine
the boundary $\partial(G,H)$ in a number of cases, highlighting properties that
might appear unexpected.
