On Chamber-regular C₂-Lattices

2025/11/11 by Franziska Stamer, Stamer, Franziska, Thomas Titz Mite +1
Mathematics · #20F65 #51E24 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2511.08312

openalex publication_date 2025/11/11 · openalex created_date 2025/11/13 · openalex updated_date 2026/07/28

Abstract

We construct the first examples of chamber-regular lattices on C₂-buildings. Assuming a conjecture of Kantor, our list of examples becomes a classification for type-preserving, chamber-regular C₂-lattices on locally finite C₂-buildings. The links of special vertices in the buildings we construct, are all isomorphic to 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 are Bruhat-Tits buildings and all our lattices are exotic building lattices.

Citations

• Property (T) for groups acting on affine buildings
• One class genera of lattice chains over number fields
• On the linearity of lattices in affine buildings and ergodicity of the singular Cartan flow
• Coarse Equivalences of Euclidean Buildings

Related

• Removing chambers in Bruhat-Tits buildings
• Non-residually finite C₂-lattices
• On Tits Buildings of Type A_n
• A criterion concerning Singer groups of generalized quadrangles, and construction of uniform lattices in \widetildeC₂-buildings
• Quotients of buildings by groups acting freely on chambers