Locally compact groups with dense orbits under Z^d-actions by automorphisms

Abstract

We consider locally compact groups G admitting a topologically transitive 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 Z^d-action and such that the 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.