A BIJECTION OF INVARIANT MEANS ON AN AMENABLE GROUP WITH THOSE ON A LATTICE SUBGROUP. Issue 2 (18th October 2021) - HOPFENSPERGER, JOHN
Abstract: Suppose G is an amenable locally compact group with lattice subgroup Γ . Grosvenor ['A relation between invariant means on Lie groups and invariant means on their discrete subgroups', Trans. Amer. Math. Soc. 288 (2) (1985), 813–825] showed that there is a natural affine injection ι : LIM(Γ )→ TLIM(G) and that ι is a surjection essentially in the case G=R^d, Γ =Z^d . In the present paper it is shown that ι is a surjection if and only if G/Γ is compact.