and Spatzier in the rank $2$ case can now be recovered as an immediate consequence of Corollary \[corollary:Epsilon\]. Actually, our proof of Theorem \[theorem:Criterion\] was inspired by their work; note however that our approach is uniform in the sense that it does not require to distinguish the cases where the diameter of $\Delta$ is even or odd. In the particular case of compact projective planes (i.e. for $m = 3$), this result was also proved by Grundhöfer in a very short and elegant way, see [@Grundhoferbis]\*[Theorem 1]{}.

\[corollary:lc\] Let $\Delta$ be a thick compact $m$-gon with $m \geq 3$. Then $\operatorname{\mathrm{Auttop}}(\Delta)$ is locally compact.

One could ask if the definition of the set $J_\varepsilon(C,C',U,U',E_0,E_1)$ whose compactness is ensured by Theorem \[theorem:Criterion\] is optimal: is there any natural bigger subset of $\operatorname{\mathrm{Auttop}}(\Delta)$ whose compactness could be proved as well for all compact polygons? In that direction, we suggest the following question; we feel that the answer is likely to be positive.

Under the same hypotheses and with the same notation as in Theorem \[theorem:Criterion\], is the following set compact? $$\tilde{J}_\varepsilon(C,C',U,U',E_0,E_1) := \left\{ \varphi \in \operatorname{\mathrm{Auttop}}(\Delta) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}\begin{array}{l}
\varphi(C) \in U, \ \varphi(C') \in U', \\
\rho(\varphi(C), \varphi(E_i)) \geq \varepsilon  \ \ 