Monday, December 2, 2013, 16.30

Seminar room 0.011, Mathematical Institute, University of Bonn

Matti Rubin: Locally moving groups and the reconstruction of structures

Abstract: Let X be a regular space and G be a group of auto-homeomorphisms of X. G is said to be a locally moving group (LM group) for X, if for every nonempty

open U X, there is $g\in G\setminus\{Id\}$ such that $G\restriction (X \setminus U) = Id$.

Let Ro(X) denote the partially ordered set of regular open subsets of X. (A subset of X is regular open, if it is equal to the interior of its closure.)

**Theorem** Let X, Y be regular spaces and G,H be LM groups for X and Y .

Suppose that $\phi$ is an isomorphism between G and H (as abstract groups).

Then there is an isomorphism $\psi$ between Ro(X) and Ro(Y ) such that for

every $g\in G$, $\phi (g) =\psi\circ g\circ \psi^{-1}$.

The above theorem has many applications in answering questions of the type:

“When does the automorphism group of a mathematical structure determine

the structure”. For example:

**Theorem** Let U, V be open subsets of normed spaces E and F. Suppose that $\phi$ is an isomorphism between the group H(X) of all auto-homeomorphisms

of X, and the group H(Y ) of all auto-homeomorphisms of Y . Then there

is a homeomorphism $\tau$ between X and Y such that for every $g\in H(X)$,

$\phi(g) =\tau\circ g\circ \tau^{-1}$.

I shall describe this framework and some results. I shall also mention some

open questions in this area.