This article is available for purchase or by subscription. See below.
Abstract

Let
$\mathbb{\mathbb{F}}$ be a field endowed
with a valuation
$v:\mathbb{\mathbb{F}}\to \mathbb{R}\cup \{\infty \}$.
When
$v$
is discrete, the classical construction of the Bruhat–Tits building
$\mathrm{\Delta}$ associated
with
${\mathrm{GL}}_{n}(\mathbb{\mathbb{F}})$ relies
on its simplicial complex structure, with vertices identified with homothety classes of
lattices in
${\mathbb{\mathbb{F}}}^{n}$.
When the valuation is not discrete (dense or surjective), the affine building
$\mathrm{\Delta}$
is no longer simplicial. We first give the construction of
$\mathrm{\Delta}$ using ultrametric
norms of
${\mathbb{\mathbb{F}}}^{n}$,
inspired by the work of Goldman and Iwahori dealing with locally compact fields
$\mathbb{\mathbb{F}}$.
This approach allows one to unify the cases where the valuation
is discrete, dense or surjective and to give a concrete model for
$\mathrm{\Delta}$.
After developing basic properties of affine buildings, we
prove the following result, in a purely geometric way. Let
$\mathrm{\Delta}$ 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
$\mathrm{\Delta}$, such that
for every isometry
$g$
of
$\mathrm{\Delta}$ and
every
$x\in \mathrm{\Delta}$, we
have
$d(x,\mathrm{Min}(g))\le Kd(x,gx)$,
where
$\mathrm{Min}(g)=\{x\in \mathrm{\Delta}:d(x,gx)\text{isminimal}\}$. In
particular
$g$
either fixes a point or translates some geodesic.
The main difficulty lies in the case where
$\mathbb{\mathbb{F}}$ 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
34.236.191.0
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
journalrecommendation form.
Or, visit our
subscription page
for instructions on purchasing a subscription.
You may also contact us at
contact@msp.org
or by using our
contact form.
Or, you may purchase this single article for
USD 40.00:
Keywords
affine buildings

Mathematical Subject Classification
Primary: 20E42, 20G25, 51E24

Milestones
Received: 16 November 2022
Accepted: 4 April 2023
Published: 13 September 2023

© 2023 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). 
