On Chamber-regular $\tilde C_2$-Lattices

TL;DR

This paper constructs the first examples of chamber-regular lattices on $ ilde C_2$-buildings, providing a classification under a conjecture and exploring unique actions on a non-Moufang generalized quadrangle.

Contribution

It introduces the first known chamber-regular lattices on $ ilde C_2$-buildings and classifies them assuming Kantor's conjecture, involving novel actions on a unique generalized quadrangle.

Findings

Constructed the first chamber-regular lattices on $ ilde C_2$-buildings.

Classified these lattices assuming Kantor's conjecture.

Identified chamber-regular actions on a non-Moufang generalized quadrangle.

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.