Infinite families of triangle presentations

TL;DR

This paper introduces the first infinite families of triangle presentations that generate lattices in exotic buildings of type _2, expanding the known examples and methods for constructing such complex geometric structures.

Contribution

It provides the first infinite families of triangle presentations leading to lattices in exotic _2 buildings of arbitrary size, and extends to other link types.

Findings

Constructed infinite families of triangle presentations for _2 buildings.

Generated lattices with arbitrarily large order in exotic buildings.

Extended methods to other link types like opposition complexes.

Abstract

A triangle presentation is a combinatorial datum that encodes the action of a group on a $2$-dimensional triangle complex with prescribed links, which is simply transitive on the vertices. We provide the first infinite family of triangle presentations that give rise to lattices in exotic buildings of type $\widetilde{\text{A}_2}$ of arbitrarily large order. Our method also gives rise to infinite families of triangle presentations for other link types, such as opposition complexes in Desarguesian projective planes.