# Isometry groups of closed Lorentz 4-manifolds are Jordan

## TL;DR

This paper proves that the isometry groups of all closed Lorentz 4-manifolds have a Jordan property, meaning their finite subgroups contain large abelian subgroups with bounded index, revealing a uniform algebraic structure.

## Contribution

It establishes the Jordan property for isometry groups of closed Lorentz 4-manifolds, a significant extension in geometric group theory and Lorentz geometry.

## Key findings

- Isometry groups of closed Lorentz 4-manifolds are Jordan groups.
- Existence of a uniform constant C bounding the index of abelian subgroups.
- Finite subgroups have large abelian subgroups with bounded index.

## Abstract

We prove that for any closed Lorentz $4$-manifold $(M,g)$ the isometry group $Isom(M,g)$ is Jordan. Namely, there exists a constant $C$ (depending on $M$ and $g$) such that any finite subgroup $\Gamma\leq Isom(M,g)$ has an abelian subgroup $A\leq\Gamma$ satisfying $[\Gamma:A]\leq C$.