Conformal group actions on Cahen-Wallach spaces

Abstract

This thesis explores the conformal structure of Cahen-Wallach spaces, and the potential construction of compact conformal quotients of Cahen-Wallach spaces. Along the way, we prove novel results about cocompact group actions, and essential homotheties. We show that any cocompact, properly discontinuous, conformal action on a Cahen-Wallach space of imaginary type must be isometric. And we demonstrate that no cocompact, properly discontinuous, conformal action on a Cahen-Wallach space can centralize an essential transformation. These results are relevant in the study of the compact Lorentzian Lichnerowicz conjecture, as they limit possible counterexamples.