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Let Γn be a polysurface group of length n. It is commensurable with a fundamental group of a 2n-manifold X(Γn) which is considered as an n-step iterated surface-fibration. Our interest is in the algebraic structure of the iterated surface-fibration. In this paper, the purpose is to find some properties of Γn independent of the choice of the filtration 1=Γ0⊂Γ1⊂Γ2⊂⋯⊂Γn. We notice that operator homomorphisms θ i : Γ i /Γ i−1 → Out (Γ i−1 ) (i=2,…,n) are of three types, and prove that the number of the operator homomorphisms of each type is independent of the choice of the filtration of Γn for some cases. Moreover, we are also concerned with the case that Γn is embedded in a connected linear Lie group without compact factor as a discrete cocompact subgroup.

The fundamental group Γ of a compact complete affine manifold is represented as an affine crystallographic subgroup of \(\). L.S. Auslander conjectured that Γ is virtually solvable. Our purpose is to find the algebraic condition on Γ which leads affirmative answer to the conjecture.

In this paper, we give a partial answer concerned with Auslander's conjecture. It is shown that every crystallographic subgroup of E( n − 2, 2) is virtually solvable.

For every affine crystallographic subgroup I? of Aff(n), L. S. Auslander conjectured that r is virtually solvable. The purpose in the present paper is to find an algebraic condition on r which leads the affirmative answer to the conjecture. The author considers the proper discontinuity of virtually torsionfree subgroups of Aff(n) by dividing into several types as for the Zariski closure of the linear part. This method leads a partial answer of Auslander's conjecture for n 5 6.