Matti Rubin: Locally moving groups and the reconstruction of structures

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.

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