Master’s thesis, which was supervised by Pierre-Emmanuel Caprace. I am very grateful to him for suggesting the topic and for his support and guidance throughout the preparation of this work. I also thank Linus Kramer for several comments on the history of compact polygons.

Preliminaries
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In this section, we give the definition and first properties of compact spherical buildings and recall what the strong transitivity and the Moufang property exactly are.

Compact spherical buildings
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Let $\Delta$ be a spherical building of rank $k$ viewed as a simplicial complex, and denote by $\Delta_r$ the set of simplices of dimension $r-1$ in $\Delta$ (for $r \in \{1, \ldots, k\}$). We fix an ordering of the $k$ types of vertices in $\Delta$. Then $\Delta$ is called a **compact spherical building** if the set $\operatorname{\mathrm{Vert}}\Delta := \Delta_1$ carries a compact topology such that the set $\operatorname{\mathrm{Cham}}\Delta := \Delta_k$ is closed in $(\operatorname{\mathrm{Vert}}\Delta)^k$ (where a chamber of $\Delta$ is viewed as a $k$-tuple of vertices, ordered according to their type). For each $r \in \{1, \ldots, k\}$, $\Delta_r$ is given the induced topology from the product topology on $(\operatorname{\mathrm{Vert}}\Delta)^r$. We also say that $\Delta$ is connected or totally disconnected when $\operatorname{\mathrm{Cham}}\Delta$ has the 