Let $G$ be a locally compact (second countable) group and let
$$
G\_0 = \cap \{ \ker\pi : \pi \text{ is a continuous finite-dimensional unitary representation of } G \}.
$$
This is the kernel of the Bohr compactification $G \to bG$, and in particular a closed normal subgroup.
Suppose that $G/G\_0$ is compact (so that $bG = G/G\_0$).


Q: Is it true that $G\_0$ is minimally almost periodic, that is, admits no non-trivial finite-dimensional unitary representations?


If $G/G\_0$ is finite, then the answer is positive, because in this case a finite-dimensional representation of $G\_0$ induces a finite-dimensional representation of $G$.



Is the kernel of the Bohr compactification minimally almost periodic provided that it is cocompact?
No, a counterexample—courtesy of Klaus Schmidt—is the semidirect product $SO(2) \ltimes \mathbb{R}^2$ taken with respect to the defining action of $SO(2)$ on $\mathbb{R}^2$.
The von Neumann kernel of this group is $\mathbb{R}^2$, which follows from the results in “The structure of homomorphisms of connected locally compact groups into compact groups” by A. I. Shtern, Izv. Ross. Akad. Nauk Ser. Mat. 75.6 (2011), pp. 195–222.


