Vol. 103, No. 2, 1982

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Locally invariant topologies on free groups

Sidney Allen Morris and Peter Robert Nickolas

Vol. 103 (1982), No. 2, 523–537

In 1948, M. I. Graev proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. In 1964, S. Świerczkowski gave a different proof, which also depends on the construction of a locally invariant topology. Yet another such construction follows from work of K. Bicknell and S. A. Morris. Graev’s topology has proved to be essential in the investigation of free products of topological groups; Świerczkowski’s topology is the key to the work of W. Taylor on varieties and homotopy laws; and Bicknell and Morris extend results of Abels on norms on free topological groups. In this paper, the three topologies are investigated in detail. It is seen that the Graev topology contains the Świerczkowski topology, which in turn contains that of Bicknell and Morris. These containments are shown to be proper in general. It is known that the topology of the free topological group is in general finer than each of these three topologies.

Mathematical Subject Classification 2000
Primary: 22A05
Secondary: 54A10
Received: 6 March 1981
Published: 1 December 1982
Sidney Allen Morris
Peter Robert Nickolas