# Differential Geometry, Lie Theory and Low-Dimensional Topology, Melbourne, Australia, 2016

Title Compact pseudo-Riemannian Einstein manifolds in low dimensions Wolfgang Globke (University of Vienna) Dr Wolfgang Globke n/a Let $M$ be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group $G$ acts transitively and isometrically on $M$. We study such spaces $M$ in the two cases where $G$ is solvable or semisimple. In the solvable case, $M$ is compact and $M=G/\Gamma$ for a lattice $\Gamma$ in $G$. Solvable Lie algebras $\mathfrak{g}$ with invariant Einstein scalar product exist only for $\dim\mathfrak{g}\leq 7$ and Witt index $\leq 2$. The existence of a compact $M$ then requires the existence of a lattice in the solvable group $G$, and in dimensions $\leq 7$, this requires $G$ to be nilpotent. We conjecture that this is true for any dimension. In fact, this holds if Schanuel's Conjecture on transcendental numbers is true. In the case where $G$ is semisimple, $M$ splits as a pseudo-Riemannian product of Einstein quotients for the compact and the non-compact factors of $G$. This is joint work with Yuri Nikolayevsky.