Let G be a group acting on either a symmetric space of noncompact type or an affine building X. I will describe some properties of the induced dynamics on the spherical building at infinity X^\infty. In many situations, G admits elements that satisfy a weak form of North-South dynamics. These elements, along with the construction of a barycenter map on triples of chambers at infinity, can help understand the C*-algebra of these groups. In particular, I will show that, even if the model space is an exotic affine building, lattices are C*-simple, and that lattices in model spaces of type A_2 (whose buildings at infinity are projective planes) are selfless. In particular, SL_3(Z) is selfless, providing the first example of a non-uniform lattice in higher rank satisfying this property. Joint work with C. Ciobotaru, I. Vigdorovich and E. Leibtag.