CoxIter – Computing invariants of hyperbolic Coxeter groups

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. Introduction LetH be the hyperbolic n-space, and let IsomH be the group of isometries ofH. For a given discrete hyperbolic Coxeter group Γ < IsomH and its associated fundamental polyhedron P ⊂ H, we are interested in geometrical and combinatorial properties of P . We want to know whether P is compact, has finite volume and, if the answer is yes, what its volume is. We also want to find the combinatorial structure of P , namely, the number of vertices, edges, 2-faces, and so on. Finally, it is interesting to find out whether Γ is arithmetic, that is, if Γ is commensurable to the reflection group of the automorphism group of a quadratic form of signature (n, 1). Most of these questions can be answered by studying finite and affine subgroups of Γ, but this involves a huge number of computations. This article presents the algorithms used in CoxIter, a computer program written in C++ designed to compute these invariants. The program is published under a free license (the GNU General Public License v3) and can be used freely in various projects. The source code and the documentation are available as supplementary material with the online version of this article and on the author’s website. The input of CoxIter is the graph of a hyperbolic Coxeter group (encoded in a simple way in a text file, see Appendix A) and a typical output can be the following. Reading file: ../graphs/14-vinb85.coxiter Number of vertices: 17 Dimension: 14 Vertices: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 Field generated by the entries of the Gram matrix: Q[sqrt(2)] File read Information Cocompact: no Finite covolume: yes Received 5 January 2015; revised 28 July 2015. 2010 Mathematics Subject Classification 5104 (primary), 52B05, 20F55 (secondary). Supported by the Schweizerischer Nationalfonds SNF no. 20002