We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups
and
. 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
.
We also give necessary and sufficient conditions for a subgroup of
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.
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