Isometry groups of closed Lorentz 4-manifolds are Jordan
It is known that every Lie group \(G\) with finite number of components has the Jordan property; that is, for some number \(C\) every nontrivial finite subgroup of \(G\) has an abelian proper subgroup of index at most \(C\). In particular, isometry groups of closed Riemannian manifolds are Jordan. However, for closed manifolds with an indefinite metric, the isometry group may have infinitely many components. For manifolds of dimension at most three, the entire diffeomorphism group is known to be Jordan. The author addresses the smallest unknown case and proves that the isometry group of a closed Lorentz 4-manifold is Jordan. The only known examples of closed 4-manifolds whose diffeomorphism group is not Jordan are the two \(S^2\)-bundles over \(T^2\). For these manifolds \(M\) the author proves that, on the other hand, for every positive number \(C\) there exists a Lorentz metric on \(M\) whose automorphism group does not satisfy the Jordan property with this choice of \(C\). So the Jordan property does not hold in a uniform way for these groups.
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