On discrete isometry groups of negative curvature
In the paper under review discrete isometry groups of a Hadamard manifold all of whose sectional curvatures are pinched by \(-1\) and \(-a^ 2\) are called negatively curved groups. The author extends several well-known results in the theory of Kleinian groups (the case of constant negative curvature) like Jørgensen inequality and Jørgensen's algebraic limit theorem to the case of negatively curved groups. One of the results is: let \(\Gamma_ n\) be a sequence of finitely generated isomorphic negatively curved nonelementary groups, acting on Hadamard spaces with uniformly pinched sectional curvatures, converging algebraically to a group \(\Gamma_ \infty\) of homeomorphisms of \(\mathbb{B}^ n\). If in addition the sequence \(\Gamma_ n\) has uniformly bounded torsion (all elements of finite order of all groups \(\Gamma_ n\) have orders bounded by the same finite constant) then the limit group \(\Gamma_ \infty\) is discrete and is isomorphic to \(\Gamma_ n\) (\(n \in \mathbb{N}\)). The condition on the torsion subgroup is essentially even in the Kleinian group case according to examples due to the author [Acta Math. 163, No. 3/4, 253-289 (1989; Zbl 0698.20037)].
discrete isometry groups of a Hadamard manifold
scientific article; zbMATH DE number 572467
scientific article