Dilations on invertible spaces

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.