#### Volume 22, issue 3 (2022)

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Geometric limits of cyclic subgroups of $\mathsf{SO}_0(1, k+1)$ and $\mathsf{SU}(1, k+1)$

### Sara Maloni and Maria Beatrice Pozzetti

Algebraic & Geometric Topology 22 (2022) 1461–1495
##### Abstract

We study geometric limits of convex-cocompact cyclic subgroups of the rank 1 groups ${SO}_{0}\left(1,k+1\right)$ and $SU\left(1,k+1\right)$. 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 ${SO}_{0}\left(1,3\right)$. We also give necessary and sufficient conditions for a subgroup of ${SO}_{0}\left(1,k+1\right)$ 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.

##### Keywords
geometric convergence, Jørgensen, parabolic, loxodromic, rank 1 Lie groups
##### Mathematical Subject Classification
Primary: 22E40
Secondary: 20F67, 30F60, 57K32