An amenable locally compact group with property (T) is compact

Given: AC, a locally compact Hausdorff group $G$ (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) with a fixed left Haar measure, and the assumptions that $G$ is amenable and has property (T).

The group is amenable in the sense of Amenable locally compact group, and under AC the Hulanicki-Reiter criterion identifies this with $1_G\prec {\lambda }_G$ (The Hulanicki–Reiter weak containment criterion for amenability).

The left regular representation ${\lambda }_G$ on $L^2(G)$ is a strongly continuous unitary representation (Left and right regular unitary representations of an LCH group). For an LCH group, $1_G\prec {\lambda }_G$ is equivalent under AC to ${\lambda }_G$ having almost invariant unit vectors (Weak containment of the trivial representation and almost invariant vectors, Almost invariant vectors for a unitary representation).

Property (T) says that every strongly continuous unitary representation with almost invariant vectors has a nonzero invariant vector (Kazhdan's property (T)).

For a fixed left Haar measure on an LCH group, a nonzero invariant vector of ${\lambda }_G$ implies that the total Haar measure is finite, and finite total Haar measure implies that $G$ is compact (Compactness, finite Haar volume and invariant vectors in the regular representation).

AC is the principle that every family of nonempty sets has a choice function (The Axiom of Choice); it is assumed in the Hulanicki-Reiter and weak-containment suppliers and in the finite-Haar-volume criterion.