Locally compact groups with dense orbits under $\bm{\mathbb{Z}}^{\bm{d}}$-actions by automorphisms

Abstract

We consider locally compact groups $G$ admitting a topologically transitive $\mathbb{Z}^d$-action by automorphisms. It is shown that such a group $G$ has a compact normal subgroup $K$ of $G$, invariant under the action, such that $G/K$ is a product of (finitely many) locally compact fields of characteristic zero; moreover, the totally disconnected fields in the decomposition can be chosen to be invariant under the $\mathbb{Z}^d$-action and such that the $\mathbb{Z}^d$-action is via scalar multiplication by non-zero elements of the field. Under the additional conditions that $G$ be finite dimensional and ‘locally finitely generated’ we conclude that $K$ as above is connected and contained in the center of $G$. We describe some examples to point out the significance of the conditions involved.