In the comments on
Is Seifert-Weber space homogeneous for a Lie group?
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (constant negative sectional curvature) then there can exist no Lie group $ G $ acting (smoothly) transitively on $ M $. Why is that?
Thoughts on the $ dim(M)=2 $ case (surface): A hyperbolic surface of finite volume must be compact. A compact hyperbolic surface must have negative Euler characteristic. But Mostow, G. D., A structure theorem for homogeneous spaces, Geom. Dedicata 114, 87-102 (2005). ZBL1086.57024 proves that a compact manifold admitting a transitive action by a Lie group must have nonnegative Euler charactersitic $ \chi \geq 0 $. Thus finite volume hyperbolic surfaces are never homogeneous.
The same Mostow paper above is used by Moishe Kohan to prove this result for 3 manifolds here
The general proof suggested here is that you show somehow that a hyperbolic manifold with a transitive $ G $ action must be simply connected and then a simply connected hyperbolic manifold is $ \mathbb{H}^n $ and so does not have finite volume.
The crux of the proof then is to show that a hyperbolic manifold with a transitive $ G $ action must be simply connected.
