Isometry Group of Lorentz Manifolds: A Coarse Perspective

TL;DR

This paper establishes a structure theorem for the isometry group of compact Lorentz manifolds with exponential growth, classifies lattices acting isometrically, and proves a Tits alternative for discrete subgroups.

Contribution

It provides a new structure theorem for isometry groups without assuming the identity component, and classifies possible lattices and discrete subgroups acting on such manifolds.

Findings

Full classification of lattices acting isometrically on compact Lorentz manifolds.

A structure theorem for isometry groups with exponential growth.

A Tits alternative for discrete subgroups of the isometry group.

Abstract

We prove a structure theorem for the isometry group Iso(M, g) of a compact Lorentz manifold, under the assumption that a closed subgroup has exponential growth. We don't assume anything about the identity component of Iso(M, g), so that our results apply for discrete isometry groups. We infer a full classification of lattices that can act isometrically on compact Lorentz manifolds. Moreover, without any growth hypothesis, we prove a Tits alternative for discrete subgroups of Iso(M, g).