Quasi-conformal rigidity of negatively curved three manifolds. (Q1811383)

This paper deals with the rigidity of 3-manifolds of infinite volume which are negatively pinched and have finitely generated fundamental groups. More precisely, let \(\widetilde M\) be a simply connected, complete Riemannian manifold with sectional curvatures ranging in the interval \([-b^2,-1]\). Consider the manifold \(M= \widetilde M_{/\Gamma}\), \(\Gamma\) being a non-elementary, torsion-free, discrete subgroup of the group \(\text{ISO}(M)\) of isometries of \(M\). The author defines infinite ends for \(M\), determining their geometrical properties. He also considers several metrics on \(S_\infty= \partial\widetilde M\), which are equivalent to Gromov's metric, and discusses suitable measures on \(S_\infty\) and the ergodicity of \(\Gamma\) with respect to them. This allows to prove the main theorem of the paper. To formulate it, we recall some basic definitions. The group \(\Gamma\) is called topologically tame if \(M= \widetilde M_{/\Gamma}\) is homeomorphic to the interior of a compact manifold with boundary. \(\Gamma\) is said to be purely loxodromic if each \(\gamma\in\Gamma\) has two fixed points on \(S_\infty\). The limit set of \(\Gamma\), denoted by \(\Lambda_\Gamma\), is the unique minimal \(\Gamma\)-invariant subset of \(S_\infty\). Denoting by \(D_\Gamma\) the positive number which represents the critical exponent of \(\Gamma\), the main result is the following: Let \(\Gamma\) be a topologically tame, purely loxodromic discrete subgroup of \(\text{ISO}(\widetilde M)\) such that \(\Lambda_\Gamma= S_\infty\). Suppose that \(f: S_\infty\to S^2\) is a quasi-conformal homeomorphism which conjugates \(\Gamma\) to a topologically tame, discrete subgroup \(\Gamma'\) of \(\text{PSL}(2, \mathbb{C})\). Then \(D_\Gamma\geq D_{\Gamma'}\). Furthermore, \(D_\Gamma= D_{\Gamma'}\) if and only if there exists \(\gamma\in \text{PSL}(2, \mathbb{C})\) such that \(\Gamma= \gamma\Gamma'\gamma^{-1}\). This theorem generalizes a rigidity theorem due to Sullivan and also a recent result of Besson, Courtois, Gallot. Further interesting consequences are stated in this paper, like the following one. Corollary: Let \(M= \widetilde M_{/\Gamma}\) be a complete 3-manifold with sectional curvatures in \([-b^2,-1]\) and \(\Gamma\) a topologically tame, pure loxodromic, with \(\Lambda_\Gamma= S_\infty\). Assume the existence of a quasi-isometric homeomorphism from \(M\) onto a hyperbolic manifold \(N\). Then \(M\) is isometric with \(N\) if and only if \(D_\Gamma= 2\) and \(\Gamma\) is Hausdorff conservative.