metric $\rho$).

![Illustration of Theorem \[theorem:Criterion\].[]{data-label="picture:Criterion"}](figure1.eps)

Remark that the conclusion of Theorem \[theorem:Criterion\] is also valid for irreducible compact spherical buildings of higher rank, but the proof that we are aware of in that case is very indirect: one deduces it from the fact that those buildings are Moufang, hence are buildings at infinity of Bruhat-Tits buildings, via the works of Tits [@Tits]\*[Addenda]{} and Grundhöfer–Kramer–Van Maldeghem–Weiss [@Grundhofer]\*[Theorem 1.1]{}.

A particular case of this criterion is the following result, which can be seen as a topological version of a well-known theorem [@Tits]\*[Theorem 4.1.1]{} of Tits. Indeed, the latter basically states that $L_0(C,C')$ (as defined below) is reduced to the identity in every thick (not necessarily topological) spherical building.

\[corollary:Epsilon\] Let $\Delta$ be a thick compact $m$-gon with $m \geq 3$ and $C, C'$ be opposite chambers of $\Delta$. There exists $\varepsilon > 0$ such that the set $$L_\varepsilon(C,C') := \left\{ \varphi \in \operatorname{\mathrm{Auttop}}(\Delta) \mid \rho(X, \varphi (X)) \leq \varepsilon \ \ \forall X \in E_1(C) \cup \{C'\}
 \right\}$$ is compact, where $\rho$ is some fixed metric associated to the topology on $\operatorname{\mathrm{Cham}}\Delta$ and $E_1(C)$ denotes the set of chambers adjacent to $C$.

We remark that the local compactness theorem of Burns 