Injective metrics on buildings and symmetric spaces

Abstract

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 ℓ ∞ $\ell ^\infty$ 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 ℓ ∞ $\ell ^\infty$ metric 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 GL ( n , R ) $\operatorname{GL}(n,\mathbb {R})$ as the space of all norms on R n $\mathbb {R}^n$ . The only exception is the special linear group: if n = 3 $n=3$ or n ⩾ 5 $n \geqslant 5$ and K $\mathbb {K}$ is a local field, we show that SL ( n , K ) $\operatorname{SL}(n,\mathbb {K})$ does not act properly and coboundedly on an injective metric space.