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Let $X$ be a non-discrete Euclidean building. Let $x \in X$, $\Delta_x$ be the germ of a Weyl-chamber based at $x$ and $\xi$ be a point at infinity. Choose $y \in \Delta_x$.

Is there an apartment containing $x$, $y$ and $\xi$?

Addendum: is there an apartment containing $x$, $\Delta_x$ and $\xi$?

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    $\begingroup$ I think you cannot ask for the apartment to contain $y$, only the germ $\Delta_x$: in a tree take a tripod with outer vertices $x$, $y$, $z$ suppose the ray from $x$ to $\xi$ passes through $z$. Then there is no apartment containing $x$, $y$, and $\xi$. $\endgroup$
    – Stefan Witzel
    Commented Jul 2, 2021 at 15:17
  • $\begingroup$ How about: take an apartment that contains $x$ and $\xi$ and then argue that the stabilizer acts sufficiently transitively to move that apartment into one that contains $\Delta_x$? If the building is a non-discrete version of Moufang, this should work. $\endgroup$
    – Stefan Witzel
    Commented Jul 5, 2021 at 20:24

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See Proposition 1.8 in Parreau's paper on non-discrete Euclidean buildings, which proves several foundational results of this nature.

https://www-fourier.ujf-grenoble.fr/~parreau/Recherche/Parreau_1999_immeubles.pdf

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