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The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3

Sira Busch

TL;DR

The paper develops a geometric, type-specific approach to the Moufang property for thick, spherical buildings by constructing explicit root elations in rank-$2$ residues for types $B_n$, $C_n$, and $H_m$ and extending these to ambient buildings when possible. It provides an alternative proof that buildings of type $B_n$ and $C_n$ are Moufang, reveals new structural features of root groups such as even self-projectivity of length $4$, and uses these ideas to show the nonexistence of thick spherical buildings of type $H_3$ and $H_4$ without relying on Tits' extension theorem. The results yield a self-contained, geometric proof framework for Moufang-ness in rank $3$, clarifying the role of projections and root elations in polar and H$_3$-type geometries. Collectively, the work offers a foundational, type-aware route toward a new proof of Tits's Moufang theorem for rank-$3$ buildings and deepens understanding of the interplay between root groups, projections, and automorphisms in these incidence geometries.

Abstract

We give direct, geometric constructions for nontrivial root elations for rank $2$ residues of higher rank buildings $Δ$ of type $\mathsf{B_n}, \mathsf{C_n}$ and $\mathsf{H_m}$ for $n \in \mathbb{N}$ and $m \in \{3,4\}$. We show that we can extend these to the ambient building in the case that $Δ$ has type $\mathsf{B_n}$ or $\mathsf{C_n}$. With that, we obtain a different proof for the fact that buildings of type $\mathsf{B_n}$ and $\mathsf{C_n}$ are Moufang. This geometric approach enables us to gain more insight into the root groups associated to these buildings and we obtain new results; Namely, that certain root elations generically fix more points than we previously knew and that every root elation in each point residual can be written as an even self-projectivity. Concerning $\mathsf{H_m}$, we will be able to see in a novel way why thick, spherical buildings of type $\mathsf{H_m}$ cannot exist. Altogther, this provides an alternative proof for the fact that all thick, irreducible, spherical buildings $Δ$ of rank 3 have the Moufang property.

The Moufang Condition and Root Automorphisms for Spherical Buildings of Rank 3

TL;DR

The paper develops a geometric, type-specific approach to the Moufang property for thick, spherical buildings by constructing explicit root elations in rank- residues for types , , and and extending these to ambient buildings when possible. It provides an alternative proof that buildings of type and are Moufang, reveals new structural features of root groups such as even self-projectivity of length , and uses these ideas to show the nonexistence of thick spherical buildings of type and without relying on Tits' extension theorem. The results yield a self-contained, geometric proof framework for Moufang-ness in rank , clarifying the role of projections and root elations in polar and H-type geometries. Collectively, the work offers a foundational, type-aware route toward a new proof of Tits's Moufang theorem for rank- buildings and deepens understanding of the interplay between root groups, projections, and automorphisms in these incidence geometries.

Abstract

We give direct, geometric constructions for nontrivial root elations for rank residues of higher rank buildings of type and for and . We show that we can extend these to the ambient building in the case that has type or . With that, we obtain a different proof for the fact that buildings of type and are Moufang. This geometric approach enables us to gain more insight into the root groups associated to these buildings and we obtain new results; Namely, that certain root elations generically fix more points than we previously knew and that every root elation in each point residual can be written as an even self-projectivity. Concerning , we will be able to see in a novel way why thick, spherical buildings of type cannot exist. Altogther, this provides an alternative proof for the fact that all thick, irreducible, spherical buildings of rank 3 have the Moufang property.

Paper Structure

This paper contains 19 sections, 20 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Buildings
  3. Point-Line Geometries
  4. Lie Incidence Geometries of Type $\mathsf{B_n}, \mathsf{C_n}$ and $\mathsf{D_n}$
  5. Apartments in polar spaces and the Moufang property
  6. Opposition, Residues and Projections
  7. Lie Incidence Geometries of Type $\mathsf{H_3}$
  8. The projection map
  9. The Moufang Property for Polar Spaces.
  10. Generalised Quadrangles in Polar Spaces are Moufang.
  11. First kind of root.
  12. Second kind of root.
  13. Polar Spaces of Rank 3 are Moufang.
  14. First kind of half-apartment.
  15. Second kind of half-apartment.
  16. ...and 4 more sections

Key Result

Lemma 1.9

Let $\Delta$ be a polar space of rank $n$. Then we can find $2n$ points $p_{-n},p_{-n+1},\ldots,p_{-1},$$p_1,p_2,\ldots,p_n$ such that $p_i\perp p_j$ if, and only if, $i+j\neq 0$, for all $i,j\in\{-n,-n+1,\ldots,-1,1,\ldots,n\}$.

Theorems & Definitions (54)

  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Remark 1.8
  • Lemma 1.9
  • proof
  • ...and 44 more