Discrete approximations for complex Kac-Moody groups

TL;DR

This paper establishes a homology equivalence between classifying spaces of discrete and topological Kac-Moody groups over different fields, extending classical results and constructing new unstable Adams operations.

Contribution

It introduces a homology equivalence for Kac-Moody groups over different fields and constructs unstable Adams operations compatible with Frobenius homomorphism.

Findings

Homology equivalence at primes q different from p

Construction of unstable Adams operations for Kac-Moody groups

New integral homology decompositions for infinite dimensional unipotent subgroups

Abstract

We construct a map from the classifying space of a discrete Kac-Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac-Moody group and prove that it is a homology equivalence at primes q different from p. This generalises a classical result of Quillen-Friedlander-Mislin for Lie groups. As an application, we construct unstable Adams operations for general Kac-Moody groups compatible with the Frobenius homomorphism. In contrast to the Lie case, the homotopy fixed points of these unstable Adams operations cannot be realized at q as the classifying spaces of Kac-Moody groups over finite fields. Our results rely on new integral homology decompositions for certain infinite dimensional unipotent subgroups of discrete Kac-Moody groups.