Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Dilations on invertible spaces

Author: Ellard Nunnally
Journal: Trans. Amer. Math. Soc. 123 (1966), 437-448
MSC: Primary 54.80
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 $ X$ is a space of the type in question and that $ G$ is an appropriate group of homeomorphisms of $ X$ onto itself. In §2 we demonstrate the existence of a nonvoid subcollection $ \mathcal{D}$, the ``topological dilations,'' of $ G$ which is characterized in Theorem 1 in the following fashion: suppose $ f \in \mathcal{D}$ and $ g \in G$, then $ g \in \mathcal{D}$ if and only if $ f$ is a $ G$-conjugate of $ g$, that is if and only if there exists an element $ h$ of $ G$ such that $ f = hg{h^{ - 1}}$.

We proceed then to show in §3 that if $ f$ and $ g$ are nonidentity elements of $ G$, then we may find $ \delta ,r \in G$ such that the product $ (rg{r^{ - 1}})(\delta f{\delta ^{ - 1}}) \in \mathcal{D}$. We then combine this fact with the characterization of $ \mathcal{D}$ mentioned above to conclude that each element of $ \mathcal{D}$ is a ``universal'' element of $ G$ in the sense that if $ d \in \mathcal{D}$, then any element $ g$ of $ G$ may be represented as the product of two $ G$-conjugates of $ d$. Furthermore we conclude that if $ g$ is not the identity element of $ G$, then $ g$ can be represented as the product of three $ G$-conjugates of any nonidentity element of $ G$.

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

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Article copyright: © Copyright 1966 American Mathematical Society