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I am trying to understand representation theory of $GL_2(F)$ where $F$ is a finite extension of $\mathbb{Q}_p$. I am using the following paper of Colmez to understand it:

https://webusers.imj-prg.fr/~pierre.colmez/kirilov.pdf

There in page 73, the author discussed briefly the Bruhat-Tits tree for $PGL_2(F)$ and define the notion of an "arete" to be "une extremite". However I am unable to visualize it and for which I am failing to understand simple proofs like the proof of Lemma III.1.13 (page 78). For example, in the proof of Lamma III.1.13, the author says:

"Soit $[r_0,r_1]$ une extremite de $\mathcal{A}$, telle que $r_o$ is not in $I_{\mathcal{A}_1 \cap \mathcal{A}_2}(W,\Pi)$; il exists .... $r_0 \in \mathcal{A}_i$"

I do not understand why such $r_0 $ should exists. I think that I am lacking a proper visualization of the Bruhat-Tits tree.

Merci to explain the visualization of the notion of concept of "extremite" in Bruhat- tits tree.

My background: I do not have a prior background of Bruhat -Tits tree or buildings, appartments in general.

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    $\begingroup$ You're aware of Serre's book? link.springer.com/book/10.1007/978-3-642-61856-7 $\endgroup$ – Ian Agol Aug 14 '16 at 18:14
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    $\begingroup$ No he doesn't define "arête" (edge) to be "extrémité" (ending edge). He says that an (oriented) "arête" is defined to be an "extrémité" if (something which amounts to saying that it source is an extremal vertex). $\endgroup$ – YCor Aug 14 '16 at 18:45

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