Abstract

Let $\mathbb{𝔽}$ be a field endowed with a valuation $v:\mathbb{𝔽}\to ℝ\cup \{\infty \}$. When $v$ is discrete, the classical construction of the Bruhat–Tits building $\mathrm{\Delta }$ associated with ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n(\mathbb{𝔽})$ relies on its simplicial complex structure, with vertices identified with homothety classes of lattices in ${\mathbb{𝔽}}^n$. 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 ${\mathbb{𝔽}}^n$, inspired by the work of Goldman and Iwahori dealing with locally compact fields $\mathbb{𝔽}$. 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 $d(x,\mathrm{Min}<!--FUNCTION\; APPLICATION-->(g))\le Kd(x,gx)$, where $\mathrm{Min}<!--FUNCTION\; APPLICATION-->(g)=\{x\in \mathrm{\Delta }:d(x,gx)\text{ is minimal}\}$. In particular $g$ either fixes a point or translates some geodesic.

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

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