The automorphism space ∑(G) of a domain without punctiform prime ends

Let G ⊆ ℂ be a simply connected domain and let ∑ (G) be its group of conformal automorphisms with the topology of uniform chordal convergence on G. In 1984 Gaier raised the question whether the connectedness of the space ∑ (G) implies that the domain G has only punctiform prime ends. As a contribution to answering this question in this paper the authors use suitable spike Junctions to construct a bounded domain without any punctiform prime end such that its automorphism space ∑ (G) is not discrete, but totally disconnected. © 2000 Mathematica Josephina, Inc.