## 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 |

DOIs | |

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

### Bibliographical note

Acknowledgements: We are grateful to Bernhard M¨uhlherr for his encouragement and for helpful conversations. We thank the editors of this special volume in honour of Jacques Tits for inviting us to contribute, and for their forbearance during the manuscript’s slow gestation. The second author was supported by a VIP grant from the Ruhr-Universit¨at Bochum. Some of this work was completed during visits to the Mathematisches Forschungsinstitut Oberwolfach: we thank them for their support. We are also indebted to the referees for their careful reading of the paper and for many suggestions making various arguments more transparent## Keywords

- math.GR
- math.MG
- 51E24, 20E42, 20G15