The Oscillator Groups,Gλ, are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric to be complete, we determine the group of isometries, we exhibit a left-invariant affine structure and prove that it is not Lorentzian. As an application, we provide ...

Let G be a locally compact amenable group, TLIM ( ) the topological left-invariant means on , and 0 limit points of Følner-nets. I show that = unless is σ -compact non-unimodular, in which case ≠ . This improves 1970 result Chou 2009 Hindman Strauss. consider analogous problem for non-topological means, give short construction net converging to invariance “weakly but not strongly,” simplifying ...