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Rigidity of limit sets for nonplanar geometrically finite Kleinian groups of the second kind

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Abstract

We consider the relation between geometrically finite groups and their limit sets in infinite-dimensional hyperbolic space. Specifically, we show that a rigidity theorem of Susskind and Swarup (Am J Math 114(2):233–250, 1992) generalizes to infinite dimensions, while a stronger rigidity theorem of Yang and Jiang (Bull Aust Math Soc 82(1):1–9, 2010) does not.

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Notes

  1. We call two groups \(G_1\) and \(G_2\) commensurable if the group \(G_1\cap G_2\) has finite index in both \(G_1\) and \(G_2\).

  2. I.e. a set for which each left coset \(g G_1\) of \(G_1\) intersects \(T\) exactly once.

  3. A group \(\Gamma \) is said to act sharply transitively on a set \(V\) if for all \(v,w\in V\), there exists a unique element \(\gamma \in \Gamma \) such that \(\gamma (v) = w\).

References

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Acknowledgments

The first-named author was supported in part by the Simons Foundation Grant #245708. The third-named author was supported in part by the NSF Grant DMS-1361677.

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Authors and Affiliations

  1. Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, TX, 76203-5017, USA

    Lior Fishman & Mariusz Urbański

  2. Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, OH, 43210-1174, USA

    David Simmons

Authors
  1. Lior Fishman
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  2. David Simmons
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  3. Mariusz Urbański
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Correspondence to Lior Fishman.

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Fishman, L., Simmons, D. & Urbański, M. Rigidity of limit sets for nonplanar geometrically finite Kleinian groups of the second kind. Geom Dedicata 178, 95–101 (2015). https://doi.org/10.1007/s10711-015-0045-0

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  • DOI: https://doi.org/10.1007/s10711-015-0045-0

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