On the operator homomorphisms of polysurface groups

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=Γ0subset ofΓ1subset ofΓ2subset ofcdots, three dots, centeredsubset ofΓn. We notice that operator homomorphisms θi : Γi/Γi−1→Out(Γi−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.