A lattice in more than two Kac--Moody groups is arithmetic

TL;DR

The paper proves that irreducible lattices in products of three or more infinite irreducible complete Kac-Moody groups are arithmetic, showing they are essentially algebraic groups over local fields, with implications for CAT(0) groups.

Contribution

It establishes that such lattices are arithmetic and identifies conditions under which they are linear or virtually simple, extending to more general CAT(0) groups.

Findings

Lattices in products of ≥3 Kac-Moody groups are arithmetic.

Irreducible lattices are either S-arithmetic or not residually finite.

Lattices are virtually simple over large fields.

Abstract

Let $\Gamma$ be an irreducible lattice in a product of n infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic group over a local field and $\Gamma$ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n is at least 2: either $\Gamma$ is an S-arithmetic (hence linear) group, or it is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.