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\vspace{2cm}





\begin{center}
\begin{large}
Kempner Colloquium
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\begin{Huge}\textsc{Locally moving groups, \\ reconstruction and undecidability\\}
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\vskip .5 in
\begin{Large}
\textbf{Mati Rubin}
\end{Large}
\vskip .1 in

(Ben Gurion University)
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\end{center}


A locally moving (LM) group for a regular topological space $X$,
is a group $G$ of homeomorphisms of $X$
which has the following property:
For every nonempty open set $U$, there is $g \in G$ such that:
(1) $g$ is not the identity function,
and (2) for every $x \in X \setminus U$, $g(x) = x$.

A group $H$ is a locally moving (LM) group, if it is isomorphic
to a group $G$ as above.
Many groups of automorphisms of topological spaces,
linear orderings, trees, Boolean algebras, measure algebras
and some other types of structures are LM.

Examples:
(1) The group of all auto-homeomorphisms of a Euclidean space is LM.
(2) The group of all $C^{\infty}$ homeomorphisms of a
��� Euclidean space is LM.
(3) The group of all automorphisms of the binary tree is LM.

Using a certain theorem about LM groups, one can prove
theorems of the following type:
If $X$ and $Y$ are topological spaces such that
$H(X)$ (= the group of all auto-homeomorphisms of $X$) is
isomorphic to $H(Y)$,�� then�� $X$ is homeomorphic to $Y$.
Similar theorems for linear orderings, trees, Boolean algebras,
measure algebras and some other types of structures are also true.

A recent theorem states that:
The first order theory of every locally moving group is undecidable.
(This solves in much more generality a question of Mark Sapir
about the R. Thompson groups).
I shall state and explain that "certain theorem" mentioned above.
Then I'll speak about consequences of that theorem including the
undecidability
theorem.

If time allows, I shall mention some open problems.

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\begin{center}
Tuesday September 9,  2014

12:10 PM - 12:50 PM

MATH 350

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