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

TL;DR

This paper demonstrates that classifying countable Tits buildings of certain types, specifically those with no rank 3 residues of spherical type and non-trivial edge labels, is as complex as classifying countable graphs, indicating maximal complexity.

Contribution

It establishes Borel completeness for the space of countable Tits buildings of specified types, extending known complexity results to these geometric structures.

Findings

Classifying countable buildings of certain types is Borel complete.

The result applies to all Coxeter diagrams with no rank 3 spherical residues and non-trivial labels.

The complexity result includes all countable generalized n-gons for n ≥ 3.

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 $n\geq 3$, the space of countable generalised $n$-gons is Borel complete.