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Affine buildings: construction by norms and study of isometries

Anne Parreau

Vol. 20 (2023), No. 2-3, 471–517

Let 𝔽 be a field endowed with a valuation v : 𝔽 {}. When v is discrete, the classical construction of the Bruhat–Tits building Δ associated with GL n(𝔽) relies on its simplicial complex structure, with vertices identified with homothety classes of lattices in 𝔽n. When the valuation is not discrete (dense or surjective), the affine building Δ is no longer simplicial. We first give the construction of Δ using ultrametric norms of 𝔽n, inspired by the work of Goldman and Iwahori dealing with locally compact fields 𝔽. This approach allows one to unify the cases where the valuation is discrete, dense or surjective and to give a concrete model for Δ.

After developing basic properties of affine buildings, we prove the following result, in a purely geometric way. Let Δ be a complete affine building, with thick spherical building at infinity and no trivial factor. There exists a constant K, depending only on the type of Δ, such that for every isometry g of Δ and every x Δ, we have d(x,Min (g)) Kd(x,gx), where Min (g) = {x Δ : d(x,gx)  is minimal}. In particular g either fixes a point or translates some geodesic.

The main difficulty lies in the case where 𝔽 is not locally compact. We give an application to nonarchimedean representations of groups with bounded generation.

Translated from the French by Harris Leung, Jeroen Schillewaert and Anne Thomas

affine buildings
Mathematical Subject Classification
Primary: 20E42, 20G25, 51E24
Received: 16 November 2022
Accepted: 4 April 2023
Published: 13 September 2023
Anne Parreau
Institut Fourier
Université Grenoble Alpes