On the operator homomorphisms of polysurface groups

Let Γ_n be a polysurface group of length n. It is commensurable with a fundamental group of a 2n-manifold X(Γ_n) 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 Γ_n independent of the choice of the filtration 1=Γ₀⊂Γ₁⊂Γ ₂⊂⊂Γ_n. 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 Γ_n for some cases. Moreover, we are also concerned with the case that Γ_n is embedded in a connected linear Lie group without compact factor as a discrete cocompact subgroup.