The Normal Subgroup Theorem for lattices on two-dimensional Euclidean buildings

TL;DR

This paper proves the normal subgroup property for groups acting on two-dimensional Euclidean buildings, showing that certain lattices are virtually simple, a first for irreducible Euclidean buildings.

Contribution

It establishes the normal subgroup property for these groups and identifies the first known simple lattices in this setting.

Findings

Every normal subgroup has finite index or is contained in the finite kernel.

Non-residually finite lattices are virtually simple.

First known simple lattices on irreducible Euclidean buildings.

Abstract

We prove the normal subgroup property for every group that acts properly and cocompactly on a two-dimensional Euclidean building: every normal subgroup has finite index or is contained in the finite kernel of the action. As a consequence, the non-residually finite lattices constructed by Titz Mite and the second author are virtually simple. They are the first known simple lattices on irreducible Euclidean buildings.