An infinitely generated intersection of geometrically finite hyperbolic groups (Q2716107)

A famous theorem of Greenberg, proved in 1960, says that if \(J\) and \(H\) are two finitely generated subgroups of a Fuchsian group, then \(J\cap H\) is finitely generated. This theorem has been generalized by several authors and in several ways for Kleinian groups in dimensions 2 and 3 and for discrete subgroups of the isometry group of the \(n\)-dimensional hyperbolic space (\(n\geq 2\)). The results in dimensions \(\geq 2\) assume geometric finiteness (which is equivalent to a finite generation in dimension 2, but which is a stronger property in higher dimensions). These results assume that the groups \(J\) and \(H\) lie inside a (larger) discrete group, and the question of whether this assumption is necessary has been an open question for a long period of time. In this paper, the author solves this question in dimensions \(\geq 4\), by giving an example of two discrete geometrically finite subgroups of the isometry group of the hyperbolic \(n\)-space (\(n\geq 4\)) whose intersection is infinitely generated (and hence not geometrically finite). The example makes use of some parabolic screw motions in the hyperbolic space which are not available in lower dimensions.