Intersections of topologically tame subgroups of Kleinian groups
Let \(\Gamma\) be a Kleinian group acting on the upper 3-space \(\mathbb{H}^3 \); we call \(\Gamma\) topologically tame if there is a finite cover of \(\Gamma \backslash \mathbb{H}^3\) which is homeomorphic to the interior of a compact 3-manifold. It is expected that all finitely generated Kleinian groups are topologically tame. Let \(\Gamma, \Gamma^*\) be Kleinian groups; we say that \(\Gamma\) is virtually fibered over \(\Gamma^*\) if there exist subgroups of finite index \(\Gamma_1, \Gamma^*_1\) of \(\Gamma, \Gamma^*\) with \(\Gamma^*_1\) a normal subgroup of \(\Gamma_1\) with quotient \(\mathbb{Z}\). In this paper the author considers under what circumstances for a topologically tame group \(\Gamma^*\) there exists an element \(\gamma\) of \(\text{PSL}_2 (\mathbb{C})\) which fixes a point of the limit set \(L(\Gamma^*)\) of \(\Gamma^*\) so that the group generated by \(\gamma\) and \(\Gamma^*\) is discrete. He proves that, essentially, either \(\gamma\) is almost an element of \(\Gamma^*\) or that any Kleinian group containing \(\gamma\) and \(\Gamma^*\) is virtually fibered over \(\Gamma^*\). The proof of this uses several geometrical and analytic tools from the theory of Kleinian groups. The author applies it to discuss \(L (\Gamma_1 \cap \Gamma_2)\) of the intersection of two topologically tame subgroups \(\Gamma_1, \Gamma_2\) of a Kleinian group \(\Gamma\) in relation to the limit sets of \(\Gamma_1\) and \(\Gamma_2\). The existence of \(\Gamma\) means that he is able to give a full description of \((L (\Gamma_1) \cap L (\Gamma_2)) - L (\Gamma_1 \cap \Gamma_2)\).
scientific article; zbMATH DE number 807401
scientific article