# The classification of $p$-compact groups for $p$ odd

### Abstract

A $p$-compact group, as defined by Dwyer and Wilkerson, is a purely homotopically defined $p$-local analog of a compact Lie group. It has long been the hope, and later the conjecture, that these objects should have a classification similar to the classification of compact Lie groups. In this paper we finish the proof of this conjecture, for $p$ an odd prime, proving that there is a one-to-one correspondence between connected $p$-compact groups and finite reflection groups over the $p$-adic integers. We do this by providing the last, and rather intricate, piece, namely that the exceptional compact Lie groups are uniquely determined as $p$-compact groups by their Weyl groups seen as finite reflection groups over the $p$-adic integers. Our approach in fact gives a largely self-contained proof of the entire classification theorem for $p$ odd.

## Authors

Kasper K. S. Andersen

Department of Mathematical Science
University of Aarhus
8000 Aarhus C
Denmark

Jesper Grodal

Department of Mathematical Science
University of Copenhagen
2100 Copenhagen
Denmark

Jesper M. Møller

Department of Mathematical Science
University of Copenhagen
2100 Copenhagen
Denmark

Antonio Viruel

Departamento Álgebra, Geometría y Topología
Universidad de Málaga
29071 Málaga
Spain