Geometric limits of cyclic subgroups of SO0 (1, k + 1) and SU (1, k + 1)

We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups SO0(1, k + 1) and SU (1, k + 1). We construct examples of sequences of subgroups of such groups that converge algebraically and whose geometric limits strictly contain the algebraic limits, thus generalizing the example first described by Jørgensen for subgroups of SO0(1, 3). We also give necessary and sufficient conditions for a subgroup of SO0(1, k + 1) to arise as the geometric limit of a sequence of cyclic subgroups. We then discuss generalizations of such examples to sequences of representations of nonabelian free groups, and applications of our constructions in that setting.