# Number of connected components of the isometry group of a simply-connected lorentzian manifold

Let $(M,g)$ be a finite-dimensional connected lorentzian manifold. Then the group $G$ of isometries of $M$ (i.e., the group of diffeomorphisms $\varphi : M \to M$ with $\varphi^* g = g$) is a Lie group. (See, e.g., the answer to this question.)

It is not difficult to construct examples of $(M,g)$ whose $G$ has infinitely many connected components, but they all share the property that the fundamental group is infinite. (See, e.g., this paper.)

Question: Suppose that $M$ is simply connected. Can $G$ still have infinitely many connected components? If so, then what about if $M$ is not just simply connected but also homogeneous?

Even when $M$ is diffeomorphic to $\mathbb{R}^2$, the isometry group can have infinitely many components. For example, the generic Lorentzian metric that is invariant under the lattice of translations $\mathbb{Z}^2\subset\mathbb{R}^2$ will have $\mathbb{Z}^2$ as its isometry group, and this has infinitely many components.
• Thanks, Robert. I was indeed to hasty in writing the question. I see how a generic lorentzian metric on the 2-torus pulls back to a metric on $\mathbb{R}^2$ with a discrete group of isometries. The case I'm really interested in is that of a homogeneous lorentzian manifold, so I look forward to any comments you can make in that situation. Cheers! – José Figueroa-O'Farrill Apr 20 '15 at 23:30