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Quasi-arithmeticity of lattices in \({{\mathrm{PO}}}(n,1)\)

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Abstract

We show that the non-arithmetic lattices in \({{\mathrm{PO}}}(n,1)\) of Belolipetsky and Thomson (Algebr Geom Topol 11(3):1455–1469, 2011), obtained as fundamental groups of closed hyperbolic manifolds with short systole, are quasi-arithmetic in the sense of Vinberg, and, by contrast, the well-known non-arithmetic lattices of Gromov and Piatetski-Shapiro are not quasi-arithmetic. A corollary of this is that there are, for all \(n\geqslant 2\), non-arithmetic lattices in \({{\mathrm{PO}}}(n,1)\) that are not commensurable with the Gromov–Piatetski-Shapiro lattices.

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Acknowledgments

The author is grateful to V. Emery for a question that led to the writing of this article, and to J. Raimbault for some helpful comments, as well as to the Max Planck Institut für Mathematik for its hospitality and financial support. The author was supported during the earlier part of this work by Swiss NSF Grant 200021_144438. Many thanks also go to the journal’s referee who offered several useful comments and references.

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  1. Max Planck Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Scott Thomson

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  1. Scott Thomson
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Correspondence to Scott Thomson.

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The author was supported during the earlier part of this work by Swiss NSF Grant 200021_144438, and is grateful to the Max Planck Institute for Mathematics (Bonn) for its hospitality and financial support during the preparation of the manuscript.

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Thomson, S. Quasi-arithmeticity of lattices in \({{\mathrm{PO}}}(n,1)\) . Geom Dedicata 180, 85–94 (2016). https://doi.org/10.1007/s10711-015-0092-6

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