In this article, we show that the Goldman-Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat-Tits buildings may be endowed with a natural piecewise $\ell^\infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural invariant Finsler metric, restricting to an $\ell^\infty$ on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. We identify the injective hull of the symmetric space of $\operatorname{GL}(n,\mathbb{R})$ as the space of all norms on $\mathbb{R}^n$. The only exception is the special linear group: if $n=3$ or $n \geq 5$ and $\mathbb{K}$ is a local field, we show that $\operatorname{SL}(n,\mathbb{K})$ does not act properly and coboundedly on an injective metric space.
Injective hulls of certain discrete metric spaces and groups
内射度量空间，或绝对1-Lipschitz收缩，与CAT（0）空间共享许多性质。20世纪60年代，J. R. Isbell证明了每个度量空间X都有一个内射壳E（X）。本文证明了：如果X是具有区间一致稳定性的连通局部有限图的顶点集，则E（X）是具有n个等距型n胞的局部有限多面体复形，对每个n，n胞等距于l^n_ \infty中的多面体.这适用于一类有限生成群G，包括所有的字双曲群和阿贝尔群等。则G通过胞腔等距作用于E（G），E（G）的第一重心重分是恰当作用的分类空间\underbar{E} G的模型.若G是双曲的，则E（G）是有限维的，作用是余紧的。特别地，每个双曲群在弱（但非粗糙）意义下在非正曲率空间上适当地和余紧地作用。
