On panel-regular ~A_2 lattices

TL;DR

This paper investigates specific lattices on ~A_2 buildings, revealing their properties can be deduced from combinatorial data, and shows the number of such lattices grows super-exponentially with the parameter q.

Contribution

It characterizes lattices on ~A_2 buildings with certain regularity and cyclic vertex stabilizers, and demonstrates their properties are determined by combinatorial data, including automorphism groups and isomorphism classes.

Findings

Number of lattices grows super-exponentially with q

Most lattices with small q are exotic

Automorphism group equals the building's automorphism group for exotic lattices

Abstract

We study lattices on ~A_2 buildings that preserve types, act regularly on each type of edge, and whose vertex stabilizers are cyclic. We show that several of their properties, such as their automorphism group and isomorphism class, can be determined from purely combinatorial data. As a consequence we can show that the number of such lattices (up to isomorphism) grows super-exponentially with the thickness parameter q. We look in more detail at the 3295 lattices with q in {2,3,4,5}. We show that with one exception for each q these are all exotic. For the exotic examples we prove that the automorphism group of the lattice and of the building coincide, and that two lattices are quasi-isometric only if they are isomorphic.