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CoxIter – Computing invariants of hyperbolic Coxeter groups

  • R. Guglielmetti (a1)

Abstract

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).

Supplementary materials are available with this article.

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References

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1. Borcherds, R., ‘Automorphism groups of Lorentzian lattices’, J. Algebra 111 (1987) 133153.
2. Bourbaki, N., Groupes et Algebres de Lie (Hermann, Paris, 1968).
3. Bugaenko, V. O., ‘Arithmetic crystallographic groups generated by reflections, and reflective hyperbolic lattices’, Lie groups, their discrete subgroups, and invariant theory , Advances in Soviet Mathematics 8 (1992) 3355.
4. Coxeter, H. S. M., ‘The complete enumeration of finite groups of the form R i 2 = (R i R j ) k ij = 1’, J. Lond. Math. Soc. 1 (1935) 2125.
5. Esselmann, F., ‘The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets’, Comment. Math. Helv. 71 (1996) 229242.
6. Felikson, A., Tumarkin, P. V. and Zehrt, T., ‘On hyperbolic Coxeter n-polytopes with n + 2 facets’, Adv. Geom. 7 (2007) 177189.
7. Hild, T., ‘Cusped hyperbolic orbifolds of minimal volume in dimensions less than 11’, PhD Thesis, University of Fribourg, 2007.
8. Johnson, N., Kellerhals, R., Ratcliffe, J. G. and Tschantz, S. T., ‘The size of a hyperbolic Coxeter simplex’, Transform. Groups 4 (1999) 329353.
9. Johnson, N., Kellerhals, R., Ratcliffe, J. G. and Tschantz, S. T., ‘Commensurability classes of hyperbolic Coxeter groups’, Linear Algebra Appl. 345 (2002) 119147.
10. Kaplinskaja, I. M. and Vinberg, E. B., ‘On the groups O 18, 1(ℤ) and O 19, 1(ℤ)’, Dokl. Akad. Nauk SSS 238 (1978) 12731275.
11. Kellerhals, R., ‘Hyperbolic orbifolds of minimal volume’, Comput. Methods Funct. Theory 14 (2014) 465481.
12. Kellerhals, R. and Perren, G., ‘On the growth of cocompact hyperbolic Coxeter groups’, European J. Combin. 32 (2011) 12991316.
13. Mcleod, J., ‘Hyperbolic reflection groups associated to the quadratic forms − 3x 0 2 + x 1 2 + ⋯ + x n 2 ’, Geom. Dedicata 152 (2011) 116.
14. Perren, G., ‘Growth of cocompact hyperbolic Coxeter groups and their rate’, PhD Thesis, University of Fribourg, 2009, http://homeweb1.unifr.ch/kellerha/pub/DissGPerren09-final.pdf.
15. Poincaré, H., ‘Sur la généralisation d’un théorème dEuler relatif aux polyèdres’, C. R. Séances Acad. Sci. 117 (1893) 144145.
16. Prokhorov, M. N., ‘The absence of discrete reflection groups with noncompact fundamental polyhedron of finite volume in Lobachevsky space of large dimension’, Izv. Math. 2 (1987) 401411.
17. Ratcliffe, J., Foundations of hyperbolic manifolds , Graduate Texts in Mathematics 149 (Springer, New York, 2006).
18. Ratcliffe, J. and Tschantz, S. T., ‘Volumes of integral congruence hyperbolic manifolds’, J. reine angew. Math. 488 (1997) 5578.
19. Ratcliffe, J. and Tschantz, S. T., ‘On volumes of hyperbolic Coxeter polytopes and quadratic forms’, J. Geom. Dedicata 163 (2013) 285299.
20. Tumarkin, P. V., ‘Hyperbolic Coxeter n-polytopes with n + 2 facets’, Math. Notes 75 (2004) 848854.
21. Vinberg, E. B., ‘On groups of unit elements of certain quadratic forms’, Sb. Math. 16 (1972) 1735.
22. Vinberg, E. B., ‘Absence of crystallographic groups of reflections in Lobachevsky spaces of large dimension’, Funct. Anal. Appl. 15 (1981) 128130.
23. Vinberg, E. B., ‘Hyperbolic reflection groups’, Russian Math. Surveys 40 (1985) 3175.
24. Vinberg, E. B., Geometry II: spaces of constant curvature , Encyclopaedia of Mathematical Sciences 29 (Springer, Berlin, 1993).
25. Vinberg, E. B., ‘Non-arithmetic hyperbolic reflection groups in higher dimensions’, Mosc. Math. J. 15 (2015) 593602.
26. Zehrt, T., ‘The covolume of discrete subgroups of Isom H2m ’, Discrete Math. 309 (2009) 22842291.
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MSC classification

Type Description Title
UNKNOWN
Supplementary materials

Guglielmetti supplementary material S1
Guglielmetti supplementary material

 Unknown (2.7 MB)
2.7 MB

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