Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T00:40:31.104Z Has data issue: false hasContentIssue false
  • English
  • Français

CoxIter – Computing invariants of hyperbolic Coxeter groups

Part of: Special aspects of infinite or finite groups Polytopes and polyhedra

Published online by Cambridge University Press:  01 December 2015

R. Guglielmetti
R. Guglielmetti*
Affiliation:
Université de Fribourg, Chemin du Musée 23, CH-1700 Fribourg, Switzerland email [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

MSC classification

Secondary: 52B05: Combinatorial properties (number of faces, shortest paths, etc.) 20F55: Reflection and Coxeter groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 18 , Issue 1 , 2015 , pp. 754 - 773
Copyright
© The Author 2015 

References

Borcherds, R., ‘Automorphism groups of Lorentzian lattices’, J. Algebra 111 (1987) 133–153.Google Scholar
Bourbaki, N., Groupes et Algebres de Lie (Hermann, Paris, 1968).Google Scholar
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) 33–55.Google Scholar
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) 21–25.Google Scholar
Esselmann, F., ‘The classification of compact hyperbolic Coxeter d-polytopes with d + 2 facets’, Comment. Math. Helv. 71 (1996) 229–242.Google Scholar
Felikson, A., Tumarkin, P. V. and Zehrt, T., ‘On hyperbolic Coxeter n-polytopes with n + 2 facets’, Adv. Geom. 7 (2007) 177–189.Google Scholar
Hild, T., ‘Cusped hyperbolic orbifolds of minimal volume in dimensions less than 11’, PhD Thesis, University of Fribourg, 2007.Google Scholar
Johnson, N., Kellerhals, R., Ratcliffe, J. G. and Tschantz, S. T., ‘The size of a hyperbolic Coxeter simplex’, Transform. Groups 4 (1999) 329–353.Google Scholar
Johnson, N., Kellerhals, R., Ratcliffe, J. G. and Tschantz, S. T., ‘Commensurability classes of hyperbolic Coxeter groups’, Linear Algebra Appl. 345 (2002) 119–147.Google Scholar
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) 1273–1275.Google Scholar
Kellerhals, R., ‘Hyperbolic orbifolds of minimal volume’, Comput. Methods Funct. Theory 14 (2014) 465–481.CrossRefGoogle Scholar
Kellerhals, R. and Perren, G., ‘On the growth of cocompact hyperbolic Coxeter groups’, European J. Combin. 32 (2011) 1299–1316.Google Scholar
Mcleod, J., ‘Hyperbolic reflection groups associated to the quadratic forms − 3x 0 2 + x 1 2 + ⋯ + x n 2 ’, Geom. Dedicata 152 (2011) 1–16.Google Scholar
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.Google Scholar
Poincaré, H., ‘Sur la généralisation d’un théorème dEuler relatif aux polyèdres’, C. R. Séances Acad. Sci. 117 (1893) 144–145.Google Scholar
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) 401–411.Google Scholar
Ratcliffe, J., Foundations of hyperbolic manifolds , Graduate Texts in Mathematics 149 (Springer, New York, 2006).Google Scholar
Ratcliffe, J. and Tschantz, S. T., ‘Volumes of integral congruence hyperbolic manifolds’, J. reine angew. Math. 488 (1997) 55–78.Google Scholar
Ratcliffe, J. and Tschantz, S. T., ‘On volumes of hyperbolic Coxeter polytopes and quadratic forms’, J. Geom. Dedicata 163 (2013) 285–299.Google Scholar
Tumarkin, P. V., ‘Hyperbolic Coxeter n-polytopes with n + 2 facets’, Math. Notes 75 (2004) 848–854.Google Scholar
Vinberg, E. B., ‘On groups of unit elements of certain quadratic forms’, Sb. Math. 16 (1972) 17–35.Google Scholar
Vinberg, E. B., ‘Absence of crystallographic groups of reflections in Lobachevsky spaces of large dimension’, Funct. Anal. Appl. 15 (1981) 128–130.Google Scholar
Vinberg, E. B., ‘Hyperbolic reflection groups’, Russian Math. Surveys 40 (1985) 31–75.Google Scholar
Vinberg, E. B., Geometry II: spaces of constant curvature , Encyclopaedia of Mathematical Sciences 29 (Springer, Berlin, 1993).Google Scholar
Vinberg, E. B., ‘Non-arithmetic hyperbolic reflection groups in higher dimensions’, Mosc. Math. J. 15 (2015) 593–602.Google Scholar
Zehrt, T., ‘The covolume of discrete subgroups of Isom H2m ’, Discrete Math. 309 (2009) 2284–2291.Google Scholar
Supplementary material: File

Guglielmetti supplementary material S1

Guglielmetti supplementary material

Download Guglielmetti supplementary material S1(File)
File 2.7 MB