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

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.

Subsection 2.B was modified and an example was added there

References in corpus (6)

• Simplicity and superrigidity of twin building lattices
• Isometry groups of non-positively curved spaces: structure theory
• Isometry groups of non-positively curved spaces: discrete subgroups
• Abstract simplicity of complete Kac-Moody groups over finite fields
• Abstract commensurators of profinite groups
• Some properties of non-positively curved lattices

Cited by in corpus (6)

• Infinite series of quaternionic 1-vertex cube complexes, the doubling construction, and explicit cubical Ramanujan complexes
• Indicability, residual finiteness, and simple subquotients of groups acting on trees
• On the structure and arithmeticity of lattice envelopes
• Bounding the covolume of lattices in products
• Amenable uniformly recurrent subgroups and lattice embeddings
• Model geometries of finitely generated groups