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On a Transformation Group

Published online by Cambridge University Press:  20 November 2018

S. K. Kaul
S. K. Kaul*
Affiliation:
University of Saskatchewan, Regina, Saskatchewan
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0. Let Γ denote a group of real linear fractional transformations (the constants defining any element of Γ are real numbers); see (3, § 2, p. 10). Then it is known that Γ is discontinuous if and only if it is discrete (3, Theorem 2F, p. 13).

Now Γ may also be regarded, equivalently, as a group of homeomorphisms of a disc D onto itself; and if Γ is discrete, then, except for elements of finite order, each element of Γ is either of type 1 or type 2 (see Definitions 0.1 and 0.2 below).

We wish to generalize the result quoted above in purely topological terms. Thus, throughout this paper we denote by X a compact metric space with metric d, and by G a topological transformation group on X each element of which, except the identity e, is either of type 1 or type 2. Let L = ﹛a ∈ X: g(a) = a for some g in G — e﹜, and . We assume furthermore that 0 is non-empty.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 935 - 941
Copyright
Copyright © Canadian Mathematical Society 1969

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