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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

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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). 
