  We study geometric limits of convex-cocompact cyclic subgroups of the rank 1
groups SO_0(1, k+1) and SU(1, k+1). We construct examples of sequences of
subgroups of such groups G that converge algebraically and whose geometric
limit strictly contains the algebraic limit, thus generalizing the example
first described by Jorgensen for subgroups of SO_0(1,3). We also give necessary
and sufficient conditions for a subgroup of SO_0(1, k+1) to arise as geometric
limit of a sequence of cyclic subgroups. We then discuss generalizations of
such examples to sequence of representations of free groups, and applications
of our constructions in that setting.
