Furstenberg boundaries for pairs of groups

Furstenberg has associated to every topological group G a universal boundary partial derivative(G). If we consider in addition a subgroup H < G, the relative notion of (G,H)-oundaries admits again a maximal object partial derivative(G, H). In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) 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 derivative(G, H). We determine the boundary partial derivative(G, H) in a number of cases, highlighting properties that might appear unexpected.