The following general question will be
considered. Suppose that $X$ and
$Y$ are topological spaces, $G$ is a subgroup of the group $H(X)$ of auto-homeomorphisms of $X$ and
$H$ is a subgroup of $H(Y)$.
Suppose further that $\phi$ is a group isomorphism between $G$ and $H$. Is there a homeomorphism
$\tau$ between $X$ and $Y$ such
that
$\phi(g) = \tau o g o \tau^{-1}$ for every $g \in G$?
The answer to the above question is
positive in several natural
situations. For example, the above is true when X and Y are open subsets of normed spaces, $G$ contains the
group $LIP^L(X)$ of locally
bilipschitz homeomorphisms of $X$ and $H$ contains $LIP^L(Y)$.
For metric spaces $X$ we consider some specific subgroups of $H(X)$. For example, let $LIP(X)$ be the group
consisting of all bilipschitz
homeomorphisms of $X$. We now ask whether the following statement is true.
If $\phi$ is an isomorphism between
the groups $LIP(X)$ and $LIP(Y)$,
then there is a bilipschitz homeomorphism $\tau$ between $X$ and $Y$ such that $\tau$ induces $\phi$.
This statement is true for bounded
open subsets of a Banach space
that have a "nice" boundary.
Another setting in which a statement similar to the above is true is for spaces which are open subsets of the
Uryson universal metric space.
