Locally compact groups with dense orbits under $\bm{\mathbb{Z}}^{\bm{d}}$-actions by automorphisms
نویسنده
چکیده
We consider locally compact groups G admitting a topologically transitive Z -action by automorphisms. It is shown that such a group G has a compact normal subgroupK ofG, invariant under the action, such thatG/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 -action and such that the Z -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.
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