Let $X$ be an affine building and $G$ a group with isometric action on $X$. For any non-empty subset $\Omega$ of $X$, we denote by $P_{\Omega}$ the fixer of $\Omega$. Similarly, for any sector $\mathfrak{C}$, we denote by $\mathfrak{B}$ the union $\bigcup_{\mathfrak{C}'\in E_{\mathfrak{C}}}P_{\mathfrak{C}'}$, where $E_{\mathfrak{C}}$ is the set of subsector of $\mathfrak{C}$. According to the third paragraph of (4.3.5) in the paper of Bruhat-Tits 1972, A decomposition $G=P_{\Omega}N\mathfrak{B}$ shows that the existence of an affine apartment $A$ such that $\Omega,\mathfrak{C}'\subset A$ for some subsector $\mathfrak{C}'$ of $\mathfrak{C}$.

I am not sure why that fact follows from the decomposition $G=P_{\Omega}N\mathfrak{B}$.

  • $\begingroup$ Could you please give the page number in Bruhat-Tits 1972 ? $\endgroup$ Feb 24, 2023 at 21:42


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