Vol. 19, No. 3, 1966

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A note on topological transformation groups with a fixed end point

William Jesse Gray

Vol. 19 (1966), No. 3, 441–447

Let (X,T,Π) be a topological transformation group, where X is a nontrivial Hausdorff continuum, and T is a topological group which leaves an endpoint e of X fixed. Wallace showed that if X is locally connected and T is cyclic, T has another fixed point. In a later paper, Wallace asked the following question: if X is a peano continuum and T is compact or abelian, does T have another fixed point?

In 1952, Wang showed that if X is arcwise connected and T is compact, T has another fixed point; Chu has recently extended this result by showing T has infinitely many fixed points. Gray has shown that in the abelian case, the answer to Wallace’s question is “no” (in general). However, if T is a generative group, and if X is arcwise connected, T has another fixed point. In this paper we will generalize the last result. In fact, we show that if X is arcwise connected or locally connected, and T is a group of the form AH, where H is a connected subgroup, and A is an abelian group generated by a compact subset, and A lies in the center of T, then T has another fixed point. We will generalize several known theorems by studying ordered spaces similar to those introduced by Wallace in 1945; in particular, we will obtain a generalized solution of the compact group problem (Theorem 2).

Mathematical Subject Classification
Primary: 54.80
Received: 24 October 1965
Revised: 5 February 1966
Published: 1 December 1966
William Jesse Gray