$\Delta$, the closure of the natural image of $\operatorname{\mathrm{Stab}}_{G}(\pi)$ in $\operatorname{\mathrm{Homeo}}(\operatorname{\mathrm{Cham}}(\pi))$ acts as a convergence group on $\operatorname{\mathrm{Cham}}(\pi)$.

The notion of a convergence group is recalled in Subsection \[subsection:convergence\] while the group of projectivities $\Pi(\pi)$ is defined in Subsection \[subsection:projectivities\]. Conditions (i) and (ii) are both independent of the group $G$. In fact, Condition (ii) in Theorem \[theorem:Characterization\] implies in particular that the closure in $\operatorname{\mathrm{Homeo}}(\operatorname{\mathrm{Cham}}(\pi))$ of the group of projectivities $\Pi(\pi)$ is a locally compact group (see Lemma \[lemma:lc\] below). In the case of *connected* compact polygons, the latter property in turn is known to be equivalent to the Moufang condition, without assuming any extra homogeneity assumption on $\Delta$ a priori (see [@Lowen]\*[Theorem 5.1]{} or [@Salzmann]\*[Theorem 66.1]{}). If one believes that the connected case is representative of the general case, one should therefore not expect Condition (ii) from Theorem \[theorem:Characterization\] to be satisfied by all infinite compact totally disconnected polygons.

We point out the following corollary of Theorem \[theorem:Characterization\].

\[corollary:affine\] Let $X$ be a locally finite thick irreducible affine building of rank $3$. If $X$ is strongly transitive, then the building at infinity $X_\infty$ is Moufang. In particular $X$ is the Bruhat–Tits building associated with a semisimple algebraic group 