Auslander's conjecture
Let
be a
discrete subgroup of the group of affine transformations
which acts discretely and properly discontinuously on
.
(Recall that
acts discretely
and properly discontinuously if for any compact
the set of
with
is
finite.)
Conjecture. (L. Auslander,1964)
If
is compact, then
is virtually solvable, that is it admits a solvable subgroup with
finite index.
Auslander's conjecture plays a central role in the study of compact
space forms
on symmetric spaces. It was studied by several distinguished authors,
including
L. Auslander, J. Milnor, G. A. Margulis, S. Friedland, Y. Kamishima, T.
Kobayashi, G. Tomanov, etc. The continuous version of Auslander's
conjecture was
proved by T. Kobayashi.
My method is to extend Radon transform to the space of bounded
distributions and
reduce the non existence of non commutative free subgroup in
to summability conditions for the Radon transform of an appropriately
chosen
bounded distribution.