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On Chamber-regular $\tilde C_2$-Lattices

Franziska Stamer, Thomas Titz Mite

TL;DR

The paper constructs the first chamber-regular lattices on $\tilde{C}_2$-buildings by exploiting chamber-regular actions on the unique generalized quadrangle $Q$ of order $(3,5)$. It develops a triangle-of-groups framework to encode global lattice actions from prescribed local actions, and uses a local-to-global CAT$(0)$ perspective to realize these actions on 2D Euclidean buildings. The authors enumerate all possible local-action configurations (via $Q$, $K_{4,4}$, and $K_{6,6}$) and classify the resulting chamber-regular lattices, obtaining $3044$ isomorphism classes; under Kantor's conjecture, these are conjectured to be the only chamber-regular lattices on locally finite $\tilde{C}_2$-buildings. The results establish the existence of exotic $\tilde{C}_2$-buildings with property $(T)$ and provide explicit presentations for the acting groups, advancing understanding of symmetry in higher-rank buildings.

Abstract

We construct the first examples of chamber-regular lattices on $\tilde C_2$-buildings. Assuming a conjecture of Kantor our list of examples becomes a classification for chamber-regular $\tilde C_2$-lattices on locally-finite $\tilde C_2$-buildings. The links of special vertices in the buildings we construct, are all isomorphic to (the incidence graph of) the unique generalized quadrangle $Q$ of order (3,5). In particular our constructions involve chamber-regular actions on $Q$. These actions on $Q$ are the first (and if Kantor's conjecture holds the only) chamber-regular actions on a finite generalized quadrangle and therefore interesting in their own right. Moreover $Q$ is not Moufang and therefore none of our examples is a Bruhat-Tits building and all our lattices are exotic building lattices.

On Chamber-regular $\tilde C_2$-Lattices

TL;DR

The paper constructs the first chamber-regular lattices on -buildings by exploiting chamber-regular actions on the unique generalized quadrangle of order . It develops a triangle-of-groups framework to encode global lattice actions from prescribed local actions, and uses a local-to-global CAT perspective to realize these actions on 2D Euclidean buildings. The authors enumerate all possible local-action configurations (via , , and ) and classify the resulting chamber-regular lattices, obtaining isomorphism classes; under Kantor's conjecture, these are conjectured to be the only chamber-regular lattices on locally finite -buildings. The results establish the existence of exotic -buildings with property and provide explicit presentations for the acting groups, advancing understanding of symmetry in higher-rank buildings.

Abstract

We construct the first examples of chamber-regular lattices on -buildings. Assuming a conjecture of Kantor our list of examples becomes a classification for chamber-regular -lattices on locally-finite -buildings. The links of special vertices in the buildings we construct, are all isomorphic to (the incidence graph of) the unique generalized quadrangle of order (3,5). In particular our constructions involve chamber-regular actions on . These actions on are the first (and if Kantor's conjecture holds the only) chamber-regular actions on a finite generalized quadrangle and therefore interesting in their own right. Moreover is not Moufang and therefore none of our examples is a Bruhat-Tits building and all our lattices are exotic building lattices.

Paper Structure

This paper contains 12 sections, 22 theorems, 14 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Triangles of groups
  3. Triangles of groups and developements
  4. Global actions with given local actions
  5. Euclidean buildings of dimension 2
  6. Definitions
  7. The local approach
  8. Recovering the action of a building lattice
  9. The generalized quadrangle of order $(3,5)$
  10. Chamber-regular actions on $\tilde{C}_2$-buildings
  11. Presentations of vertex groups
  12. Actions on the edge groups

Key Result

Theorem 1

There are exactly 3044 chamber-regular lattices (up to isomorphism) on $\tilde{C}_2$-buildings whose links of special vertices are the generalized quadrangle of order (3,5).

Figures (5)

  • Figure 1: A triangle of groups.
  • Figure 2: The triangle of groups $T((\gamma_{ij})_{ij})$.
  • Figure 3: The configuration of $x_1, x_2, x_3$ in the proof of Proposition \ref{['prop:recover']} in type $\tilde{C}_2$.
  • Figure 4: The generalized quadrangle of order (5,3), the dual of $Q$.
  • Figure 5: The triangle of groups $\textbf{T} (V_1, V_2, V_3, E_1, E_2, E_3, (\varepsilon_{ij})_{ij})$.

Theorems & Definitions (49)

  • Theorem 1: Main Theorem
  • Definition 2
  • Remark 3
  • Definition 4
  • Definition 5
  • Remark 6
  • Definition 7
  • Definition 8
  • Proposition 10
  • Definition 11
  • ...and 39 more