Auslander's conjecture

Let Gamma be a discrete subgroup of the group of affine transformations Aff(R^n) which acts discretely and properly discontinuously on R^n. (Recall that Gamma acts discretely and properly discontinuously if for any compact C \in R^n the set of gamma \in Gamma with gamma(K) \cap K != emptyis finite.)

Conjecture. (L. Auslander,1964)
If R^n / Gamma is compact, then Gamma 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 Gamma to summability conditions for the Radon transform of an appropriately chosen bounded distribution.

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