## Abstract

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 VG(R) which is an analogue of the vector building for a semisimple group; we call VG(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 VG(R) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, VG(R) becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.

∆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 VG(R) which is an analogue of the vector building for a semisimple group; we call VG(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 VG(R) and show they are all bi-Lipschitz equivalent to each other; with this extra structure, VG(R) becomes a complete metric space. Finally, we present some motivation in terms of geometric invariant theory and variations on the Tits Centre Conjecture.

Original language | English |
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Journal | Innovations in Incidence Geometry: Algebraic, Topological and Combinatorial |

Publication status | Accepted/In press - 19 May 2023 |