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

Geometric limits of cyclic subgroups of SO_0(1, k+1) and SU(1, k+1)