Local compactness in the automorphism space ∑(G)

Let be a simply-connected domain and let ∑(G) be its group of conformal automorphisms with the topology of uniform chordal convergence on G. The space ∑(G) can never be compact, but in 1984 D. Gaier could give some examples of at least locally compact spaces ∑(G) (cf. Math. Z. 187 (1984), 227–257). Therefore he asked whether each space ∑(G) is locally compact. The answer, however, is negative because G. Schmieder (Math. Z. 209 (1992), 245–249) and the author have constructed several examples of domains with non-locally compact spaces ∑(G). In this paper we give a complete description of the topological types of all locally compact automorphism spaces ∑(G).