Topology and dynamics of compact plane waves

Malek Hanounah; Ines Kath; Lilia Mehidi; Abdelghani Zeghib

doi:10.1515/crelle-2024-0095

Article

Topology and dynamics of compact plane waves

Malek Hanounah , Ines Kath , Lilia Mehidi and Abdelghani Zeghib

Published/Copyright: January 8, 2025

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2025 Issue 820

Abstract

We study compact locally homogeneous plane waves. Such a manifold is a quotient of a homogeneous plane wave 𝑋 by a discrete subgroup of its isometry group. This quotient is called standard if the discrete subgroup is contained in a connected subgroup of the isometry group that acts properly cocompactly on 𝑋. We show that compact quotients of homogeneous plane waves are “essentially” standard; more precisely, we show that they are standard or “semi-standard”. We find conditions which ensure that a quotient is not only semi-standard but even standard. As a consequence of these results, we obtain that the flow of the parallel lightlike vector field of a compact locally homogeneous plane wave is equicontinuous.

A Cocompact proper actions of Lie groups

In this appendix, we consider cocompact proper actions of Lie groups admitting a torsion-free uniform lattice.

Proposition A.1

Let 𝐺 be a connected Lie group which admits a torsion-free uniform lattice Γ. Assume that 𝐺 acts properly cocompactly on a contractible manifold 𝑋. Then 𝐺 acts transitively.

The proof uses techniques from cohomology theory of discrete groups.

Proof

Since Γ is torsion-free, Γ \ X is a closed K ⁢ ( Γ , 1 ) manifold. We have by [7, VIII, (8.1)] that the cohomological dimension of Γ is equal to dim ( Γ \ X ) . Now, let 𝐾 be a maximal compact subgroup of 𝐺. We know that 𝐺 is diffeomorphic to K × R k ; see [22]. We claim that Γ \ G / K is a K ⁢ ( Γ , 1 ) manifold. Indeed, Γ acts freely (torsion-free) and cocompactly. Moreover, Γ acts properly, since the fiber bundle map π : G → G / K is a proper map. Hence [7] we get that dim ( Γ \ G / K ) = dim ( Γ \ X ) , i.e. dim ( G ) = dim ( K ) + dim ( X ) . Because the 𝐺-action is proper (in particular, the stabilizer of any point is compact), we conclude that the 𝐺-orbits are open. The claim follows from the connectedness of 𝑋. ∎

When 𝐺 is linear, Selberg’s lemma applies. Namely, any finitely generated subgroup of 𝐺 is virtually torsion-free. For linear groups, we get the following corollary.

Corollary A.2

Let 𝐺 be a connected linear Lie group which admits a uniform lattice Γ. Assume that 𝐺 acts properly cocompactly on a contractible manifold 𝑋. Then 𝐺 acts transitively.

Proposition A.3

G ρ is linear.

Proof

We have a natural morphism

f : G ρ → Aut ⁢ ( Heis ) ⋉ Heis , f ⁢ ( r , k , h ) = ( ρ ⁢ ( r , k ) , h )

(which is not necessarily injective). Moreover, the group Aut ⁢ ( Heis ) ⋉ Heis is linear. Indeed, it has a trivial center (due to existence of homotheties); hence the adjoint representation

Ad : Aut ⁢ ( Heis ) ⋉ Heis → GL ⁢ ( g ) ,

where 𝔤 is the Lie algebra of Aut ⁢ ( Heis ) ⋉ Heis , is an embedding (faithful). Define now

Φ : G ρ → ( R × K ) × GL ⁢ ( g ) , Φ ⁢ ( r , k , h ) = ( r , k , Ad ⁢ ( f ⁢ ( r , k , h ) ) ) .

Then Φ is clearly a faithful morphism into ( R × K ) × GL ⁢ ( g ) , which is a linear group. The claim follows. ∎

B A non-periodic example

Let L = R ⋉ R n + 1 , with coordinates ( v , y ) ∈ R n + 1 = R × R n and the ℝ-action defined by t ⋅ ( v , y ) = ( v , R t ⁢ ( y ) ) , where R t is some periodic elliptic action on R n . Define a Lorentzian left-invariant metric 𝑔 on 𝐿 such that the induced metric on R n + 1 is degenerate and V : = ∂ v is lightlike. Then ( L , g ) is a homogeneous plane wave, by [15, Theorem 3]. Let Γ : = ⟨ γ ̂ ⟩ × Γ 0 , with Γ 0 a lattice in R n + 1 and γ ̂ generates a lattice in the ℝ-factor acting trivially on Γ 0 . Suppose further that Γ 0 does not intersect the subgroup generated by the 𝑣-translations. Then Γ is a (uniform) lattice in 𝐿, and the flow of 𝑉 is not periodic in Γ \ L .

Acknowledgements

We thank the referee for the valuable comments and suggestions that helped improve the quality of the presentation.

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Received: 2023-10-01

Revised: 2024-11-08

Published Online: 2025-01-08

Published in Print: 2025-03-01

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