Reduction to the Bounded Image Theorem

Abstract

Theorems 15.5 and 15.7 follow directly from the Maskit combination theorems (Theorems 4.101 and 4.103).

Now we shall give yet another reformulation of the generic case of Theorem 15.4 using the language of Teichmüller theory. We recall that there is a natural embedding $\alpha\,:\,\mathcal{T}_{{\sum}}(G) \hookrightarrow \mathcal{T}(F_1) \times \mathcal{T}(F_2)$; see Section 8.11.

Let c _j denote the projections from $\mathcal{T}_{{\sum}}(G)\ {\rm to}\ \mathcal{T}(F_j) (j = 1,2)$. The gluing homeomorphism $\tau\ {\rm of}\ \sum \subset \partial_0 N$ reverses the induced orientation of the boundary. Consider the product manifold $\dot{M}(F_1) \sqcup \dot{M}(F_1) \cong [-1, 1] \times \sum$, where we identify $\{+1\} \times \sum$ with $\Omega_1/F_1 \cup \Omega2/F_2; \Omega_j$ is contained in the domain of discontinuity of $\Omega(G_j)$ if N is not connected and in $\Omega(G)$ if N is connected.