Conformal quotients of plane waves, and Lichnerowicz conjecture in a locally homogeneous setting
Lilia Mehidi
TL;DR
This work analyzes conformal actions by similarity groups on simply connected, non-flat homogeneous plane waves, showing that proper cocompact similarity actions can be non-isometric, unlike Riemannian cases. It develops a reduction framework to study compact quotients via $(G,X)$-manifolds and homogeneous plane waves, proving a Lorentzian Lichnerowicz-type result in the locally homogeneous setting. The authors classify discrete cocompact actions, construct explicit non-isometric examples, and analyze normalizers to establish that the conformal group of any compact quotient is non-essential, either by remaining within isometries or by preserving a conformal Lorentz metric. The results illuminate the richer structure of plane-wave quotients, raise Fried-type questions for incomplete models, and advance understanding of essentiality in Lorentzian conformal geometry with potential impact on conformal and geometric group theory.
Abstract
In the first part of the paper, we study conformal groups that act properly discontinuously and cocompactly on simply connected, non-flat homogeneous plane waves. We show that proper cocompact similarity actions that are not isometric can occur, in contrast to the behavior of Riemannian and Lorentzian affine similarity actions. In the second part, we consider the Lorentzian conformal Lichnerowicz conjecture, which states that if the conformal group of a compact Lorentzian manifold acts without preserving any metric in the conformal class, then the manifold must be conformally flat. We prove the conjecture in a locally homogeneous setting.