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 X is a space of the type in question and that G G is an appropriate group of homeomorphisms of X X onto itself. In §2 we demonstrate the existence of a nonvoid subcollection D \mathcal {D} , the “topological dilations,” of G G which is characterized in Theorem 1 in the following fashion: suppose f ∈ D f \in \mathcal {D} and g ∈ G g \in G , then g ∈ D g \in \mathcal {D} if and only if f f is a G G -conjugate of g g , that is if and only if there exists an element h h of G G such that f = h g h − 1 f = hg{h^{ - 1}} . We proceed then to show in §3 that if f f and g g are nonidentity elements of G G , then we may find δ , r ∈ G \delta ,r \in G such that the product ( r g r − 1 ) ( δ f δ − 1 ) ∈ D (rg{r^{ - 1}})(\delta f{\delta ^{ - 1}}) \in \mathcal {D} . We then combine this fact with the characterization of D \mathcal {D} mentioned above to conclude that each element of D \mathcal {D} is a “universal” element of G G in the sense that if d ∈ D d \in \mathcal {D} , then any element g g of G G may be represented as the product of two G G -conjugates of d d . Furthermore we conclude that if g g is not the identity element of G G , then g g can be represented as the product of three G G -conjugates of any nonidentity element of G G . Finally, we apply the conclusions to groups of homeomorphisms of certain spaces: for example spheres, cells, the Cantor set, etc.