Geometric limits of cyclic subgroups of SO0(1,k + 1) and SU(1,k + 1)

Maloni, Sara; Pozzetti, Maria Beatrice

doi:10.2140/agt.2022.22.1461

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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

DOI: 10.2140/agt.2022.22.1461

Abstract

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

Keywords

geometric convergence, Jørgensen, parabolic, loxodromic, rank 1 Lie groups

Mathematical Subject Classification

Primary: 22E40

Secondary: 20F67, 30F60, 57K32

References

Publication

Received: 28 December 2020

Revised: 9 March 2021

Accepted: 5 April 2021

Published: 25 August 2022

Authors

Sara Maloni
Department of Mathematics University of Virginia Charlottesville, VA United States
https://sites.google.com/view/sara-maloni
Maria Beatrice Pozzetti
Department of Mathematics University of Heidelberg Heidelberg Germany
http://www.mathi.uni-heidelberg.de/~pozzetti

Volume 22, issue 3 (2022)