Injective metrics on buildings and symmetric spaces

In this article, we show that the Goldman–Iwahori metric on space of all norms a fixed vector satisfies Helly property for balls. On non-Archimedean side, deduce most classical Bruhat–Tits buildings may be endowed with natural piecewise ? ? $\ell ^\infty$ which is injective. We also prove semisimple groups over local fields act properly and cocompactly graphs. This gives another proof biautomaticity their uniform lattices. Archimedean symmetric spaces non-compact type invariant Finsler metric, restricting to an each flat, coarsely injective spaces. identify hull GL ( n , R ) $\operatorname{GL}(n,\mathbb {R})$ as $\mathbb {R}^n$ . The only exception special linear group: if = 3 $n=3$ or ? 5 $n \geqslant 5$ K {K}$ field, SL $\operatorname{SL}(n,\mathbb {K})$ does not coboundedly space.