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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

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.

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geometric convergence, Jørgensen, parabolic, loxodromic, rank 1 Lie groups
Mathematical Subject Classification
Primary: 22E40
Secondary: 20F67, 30F60, 57K32
Received: 28 December 2020
Revised: 9 March 2021
Accepted: 5 April 2021
Published: 25 August 2022
Sara Maloni
Department of Mathematics
University of Virginia
Charlottesville, VA
United States
Maria Beatrice Pozzetti
Department of Mathematics
University of Heidelberg