Discrete approximations for complex Kac-Moody groups

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.

New title and revised introduction, references added; results and proofs unchanged, 31 pages, 1 figure

References in corpus (2)

• Kac-Moody groups as discrete groups
• Cohomology of Kac-Moody groups over a finite field

Cited by in corpus (1)

• Recognizing nullhomotopic maps into the classifying space of a Kac-Moody group