Dilations on invertible spaces

Author:
Ellard Nunnally

Journal:
Trans. Amer. Math. Soc. **123** (1966), 437-448

MSC:
Primary 54.80

DOI:
https://doi.org/10.1090/S0002-9947-1966-0208575-7

MathSciNet review:
0208575

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Abstract: This paper primarily concerns certain groups of homeomorphisms which are associated in a natural way with a variety of spaces, which satisfy a set of axiomatic conditions put forth in §1.

Let us suppose that is a space of the type in question and that is an appropriate group of homeomorphisms of onto itself. In §2 we demonstrate the existence of a nonvoid subcollection , the ``topological dilations,'' of which is characterized in Theorem 1 in the following fashion: suppose and , then if and only if is a -conjugate of , that is if and only if there exists an element of such that .

We proceed then to show in §3 that if and are nonidentity elements of , then we may find such that the product . We then combine this fact with the characterization of mentioned above to conclude that each element of is a ``universal'' element of in the sense that if , then any element of may be represented as the product of two -conjugates of . Furthermore we conclude that if is not the identity element of , then can be represented as the product of three -conjugates of *any* nonidentity element of .

Finally, we apply the conclusions to groups of homeomorphisms of certain spaces: for example spheres, cells, the Cantor set, etc.

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DOI:
https://doi.org/10.1090/S0002-9947-1966-0208575-7

Article copyright:
© Copyright 1966
American Mathematical Society