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On commensurable hyperbolic Coxeter groups

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Abstract

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space \(\mathbb H^n\), new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions.

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Acknowledgments

The authors would like to thank Vincent Emery and John Ratcliffe for helpful discussions. The first author was fully and the second and third authors were partially supported by Schweizerischer Nationalfonds 200020-144438 and 200020-156104.

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Authors and Affiliations

  1. Department of Mathematics, University of Fribourg, 1700, Fribourg, Switzerland

    Rafael Guglielmetti, Matthieu Jacquemet & Ruth Kellerhals

Authors
  1. Rafael Guglielmetti
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  2. Matthieu Jacquemet
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  3. Ruth Kellerhals
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Correspondence to Ruth Kellerhals.

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In memoriam Colin Maclachlan.

Appendix: Tumarkin’s Coxeter pyramid groups

Appendix: Tumarkin’s Coxeter pyramid groups

The classification of the cofinite Coxeter groups in \(\text {Isom}(\mathbb H^n)\) of rank \(n+2\) whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions is due to Tumarkin [29, 30]. The results are summarised in the Figs. 19–21 (see [30, Section 4]).

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Guglielmetti, R., Jacquemet, M. & Kellerhals, R. On commensurable hyperbolic Coxeter groups. Geom Dedicata 183, 143–167 (2016). https://doi.org/10.1007/s10711-016-0151-7

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