\pi$, $g(\pi') = \pi'$ and $g(D_1) = D_2$.

The Moufang property is another transitivity hypothesis on the automorphism group of a building. We say that a thick irreducible spherical building $\Delta$ is **Moufang** (or satisfies the **Moufang property**) if for each root $\alpha$ of $\Delta$, the **root group** $$U_\alpha := \{g \in \operatorname{\mathrm{Aut}}(\Delta) \mid g \text{ fixes every chamber having a panel in } \alpha \setminus \partial\alpha\}$$ acts transitively on the set of apartments of $\Delta$ containing $\alpha$. A natural wish would be to replace $\operatorname{\mathrm{Aut}}(\Delta)$ with $\operatorname{\mathrm{Auttop}}(\Delta)$ in the case where $\Delta$ is a compact building, but this is actually not necessary. Indeed, one can show that every element of a root group is automatically continuous, hence contained in $\operatorname{\mathrm{Auttop}}(\Delta)$ (see [@Grundhofer]\*[Corollary 6.17]{}).

A compactness criterion for subsets of Auttop(Delta) {#section:criterion}
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This section is dedicated to the proof of Theorem \[theorem:Criterion\] which gives, in a compact polygon $\Delta$, a sufficient condition on subsets of $\operatorname{\mathrm{Auttop}}(\Delta)$ for being compact.

Relative compactness in Auttop(Delta)
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To prove Theorem \[theorem:Criterion\], we need some characterization of relative compactness in $\operatorname{\mathrm{Auttop}}(\Delta)$. The tool that will help us is the Arzela-Ascoli theorem. For the reader’s convenience, we recall its statement below. First recall that 