On Ahlfors' weak finiteness theorem (Q1070079)

For all \(n\geq 3\), Ahlfors' weak finiteness theorem in hyperbolic space \(H^ n\) asserts the finiteness of the dimension of a certain space Q(\(\Gamma)\) obtained from quasi-conformal deformations which are automorphic with respect to a group \(\Gamma\) of isometries acting discontinuously on \(H^ n\) under the assumption that \(\Gamma\) is finitely-generated [see \textit{L. V. Ahlfors}, ''Möbius transformations in several dimensions'' (1981; Zbl 0517.30001)]. In the case \(n=3\), this was important in Ahlfors' finiteness theorem and in the case \(n=2\), \(Q(\Gamma)\) corresponds to the bounded integrable automorphic forms of weight -4 with extension across \(\Omega(\Gamma)\cap {\mathbb{R}}\) if it is non-empty. Here the author introduces a space \(\tilde Q(\Gamma)\) also obtained from quasi-conformal deformations with \(Q(\Gamma)\subset \tilde Q(\Gamma)\). By showing that \(\tilde Q(\Gamma)\) is the image of a space \(P(\Gamma)\) consisting of certain quasi-conformal deformations of the boundary \({\hat {\mathbb{R}}}^{n-1}\) of \(H^ n\) and that \(P(\Gamma)\) is finite-dimensional the author deduces Ahlfors' weak finiteness theorem. In addition in the case \(n=3\) he deduces that \(\tilde Q(\Gamma)=Q(\Gamma)=\{0\}\).