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

Gianluca Paolini, Davide Emilio Quadrellaro

TL;DR

The paper addresses the anti-classification problem for countable Tits buildings of a fixed Coxeter type D with no spherical rank-3 residues. It develops a two-stage method: encode countable trees into countable generalized n-gons via a finitary confined-pattern construction and a Borel reduction, then extend to arbitrary buildings using Ronan’s free completion built from these polygons. The main result is that the space of countable buildings of such type D is Borel complete, matching the complexity of countable graphs; in particular, the space of countable generalized n-gons is Borel complete for every n≥3. This generalizes prior Borel completeness results for projective planes and highlights the robust anti-classification phenomenon in incidence-geometric structures.

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.

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

TL;DR

The paper addresses the anti-classification problem for countable Tits buildings of a fixed Coxeter type D with no spherical rank-3 residues. It develops a two-stage method: encode countable trees into countable generalized n-gons via a finitary confined-pattern construction and a Borel reduction, then extend to arbitrary buildings using Ronan’s free completion built from these polygons. The main result is that the space of countable buildings of such type D is Borel complete, matching the complexity of countable graphs; in particular, the space of countable generalized n-gons is Borel complete for every n≥3. This generalizes prior Borel completeness results for projective planes and highlights the robust anti-classification phenomenon in incidence-geometric structures.

Abstract

We prove that, for every Coxeter diagram with no rank residues of spherical type and such that has not only edges labelled by , the space of countable (Tits) buildings of type is Borel complete, that is, classifying countable buildings of type up to isomorphism is as hard as classifying countable graphs up to isomorphism. In particular, for every , the space of countable generalised -gons is Borel complete.
Paper Structure (8 sections, 6 theorems, 8 equations, 4 figures)

This paper contains 8 sections, 6 theorems, 8 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalised polygons
  4. Chambers complexes and buildings
  5. Generalised polygons
  6. The finitary construction
  7. The Borel reduction
  8. Buildings

Key Result

Lemma 2.9

Let $A$ be a partial confined generalised $n$-gon. If $B \subseteq F(A)$ (where $F(A)$ is as in def_free_completion) is finite and confined, then $B \subseteq A$.

Figures (4)

  • Figure 1: The Heawood Graph.
  • Figure 2: The Tutte-Coxeter Graph.
  • Figure 3: The configuration $A$ for $n\geqslant 5$ odd.
  • Figure 4: The configuration $A$ for $n\geqslant 6$ even.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2: Generalised $n$-gon
  • Definition 2.3
  • Remark 2.4
  • Definition 2.6
  • Definition 2.7: Confined generalised $n$-gon
  • Remark 2.8
  • Lemma 2.9
  • proof
  • Definition 2.10: Simplicial complexes
  • ...and 23 more