has attracted much attention; in some sense, this is the topic of the encyclopaedic book [@Salzmann], mainly devoted to the study of *connected* compact projective planes. The relevance of that problem is also well illustrated by the work of Burns and Spatzier: a key result from [@Burns] that is used in the proof of Rank Rigidity in [@Burns2] asserts that a thick irreducible *connected* compact spherical building of rank $\geq 2$ is the spherical building associated with a simple Lie group if and only if its automorphism group is strongly transitive (see Subsection 2.2 for the definition of strong transitivity). This result has been improved by Grundhöfer, Knarr and Kramer who showed in [@GKK] and [@GKK2] that the same classification is true for connected buildings admitting a chamber transitive automorphism group. The starting point of the present article is the question whether such a statement could hold beyond the connected case. A precise formulation can be stated as follows.

Let $\Delta$ be an infinite thick irreducible compact spherical building of rank at least $2$. Then $\Delta$ is the spherical building associated with a semisimple algebraic group over a non-discrete locally compact field if and only if $\Delta$ is strongly transitive.

