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 $\Gamma $ . 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 $\iota : {\text {LIM}}(\Gamma )\to {\text {TLIM}}(G)$ and that $\iota $ is a surjection essentially in the case $G={\mathbb R}^d$, $\Gamma ={\mathbb Z}^d$ . In the present paper it is shown that $\iota $ is a surjection if and only if $G/\Gamma $ is compact.