Download this article
 Download this article For screen
For printing
Recent Issues
Volume 18
Volume 16
Volume 15
Volume 14
Volume 13
Volume 12
Volume 11
Volume 10
Volume 9
Volume 8
Volume 6+7
Volume 5
Volume 4
Volume 3
Volume 2
Volume 1
The Journal
About the journal
Ethics and policies
Peer-review process
Submission guidelines
Submission form
Editorial board
ISSN (electronic): 2640-7345
ISSN (print): 2640-7337
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
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

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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