Borel completeness of Tits buildings with no rank 3 residues of spherical type

Abstract

We prove that, for every Coxeter diagram $D$ with no rank $3$ residues of spherical type and such that $D$ has not only edges labelled by $2$, the space of countable (Tits) buildings of type $D$ is Borel complete, that is, classifying countable buildings of type $D$ up to isomorphism is as hard as classifying countable graphs up to isomorphism. In particular, for every $n\ge 3$, the space of countable generalised $n$-gons is Borel complete.