Furstenberg boundaries for pairs of groups

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.