Homogeneous orbit closures and Diophantine approximations of algebraic numbers

Abstract:

The content of the talk is a joint work with Elon Lindenstrauss.

In homogeneous dynamics one studies the action of a group A on the
homogeneous space G/H. Usually G is a Lie group and A,H are closed
subgroups of G (in our discussion H will be a lattice in G). I will be
interested in my talk in points, x, of G/H for which the closure of
the orbit Ax is itself an orbit of a (possibly bigger) group A'<G.
Such a point is called "A-regular". In the case A is generated by
unipotent subgroups, Ratner has shown that any point is A-regular. In
the case of A being a diagonalizable group (the opposite extreme of
unipotent) this is known not to be true but still, in some cases we
expect some sort of this "rigidity".

I will describe some specific examples of A-regular points, where
G=SL(n,R), H=SL(n,Z) and A= the group of diagonal matrices in G (n>2).
These examples are connected to number theory, and as an application
of the A-regularity we obtain some new information on the Diophantine
properties of algebraic numbers.

I will NOT assume the audience is familiar with the subject hence
everyone is welcome.

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