Edifices: building-like spaces associated to linear algebraic groups

Given a semisimple linear algebraic k-group G, one has a spherical building ∆ G , and one can interpret the geometric realisation ∆ G (R) of ∆ G in terms of cocharacters of G.The aim of this paper is to extend this construction to the case when G is an arbitrary connected linear algebraic group; we call the resulting object ∆ G (R) the spherical edifice of G.We also define an object V G (R) which is an analogue of the vector building for a semisimple group; we call V G (R) the vector edifice.The notions of a linear map and an isomorphism between edifices are introduced; we construct some linear maps arising from natural group-theoretic operations.We also devise a family of metrics on V G (R) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, V G (R) becomes a complete metric space.Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.