Coxiter

• Euler characteristic (and covolume for even dimensions)
• f-vector (face count vector of the associated polyhedron)
• Cocompactness (whether the quotient is compact)
• Finite covolume (whether the group has finite covolume)
• Growth series and growth rate
• Arithmeticity (whether the group is arithmetic, in the non-cocompact case)

Mathematical Background

A Coxeter group is a group generated by reflections, defined by a Coxeter matrix $(m_{ij})$ where:

• $m_{ii} = 1$
• $m_{ij} = m_{ji} \in {2, 3, \ldots, \infty}$ for $i \neq j$

The Coxeter graph (or Coxeter diagram) represents this visually:

• Vertices correspond to generators/reflection hyperplanes
• An edge between vertices $i$ and $j$ labeled $m_{ij}$ (often omitted when $m_{ij} = 2$)
• No edge means $m_{ij} = 2$ (commuting generators)
• Weight $\infty$ represented as 0 in input files

Hyperbolic Coxeter groups act on hyperbolic space $H^n$ with:

• Fundamental domain a polyhedron (possibly with vertices at infinity)
• Reflections in hyperplanes bounding the polyhedron

Input File Format

CoxIter reads Coxeter groups from text files with the following format:

d n                                    # Line 1: vertices, dimension
vertices labels: v1 v2 v3 ...         # Line 2 (optional): custom labels
v_i v_j weight                        # Lines 3+: edges with weights

Format Details

Line 1 (mandatory): d n

• d: number of vertices (generators/reflection hyperplanes)
• n: dimension of hyperbolic space (optional but recommended)
  • If omitted, CoxIter determines dimension by looking at maximal spherical and euclidean subgraphs
  • Important: This requires the polyhedron to have at least one vertex. If unsure whether this holds, specify the dimension.

Line 2 (optional): vertices labels: label1 label2 ...

• Custom labels for vertices (letters, digits, -, _ allowed; no spaces)
• If omitted, vertices are labeled 1, 2, ..., d

Edge lines: vertex1 vertex2 weight

• Remarks:
  1. 0 is used to specify the weight infinity; 1 is used to specify a dotted edge
  2. Only one line per edge is sufficient (undirected - order doesn't matter)
  3. The weight 2 doesn't have to be specified (no edge means weight 2, i.e., orthogonal hyperplanes)
• Weights encode the Coxeter matrix entries:
  • 0 = weight infinity (hyperplanes parallel at infinity)
  • 1 = dotted edge (hyperplanes ultra-parallel)
  • 3, 4, 5, ... = angle $\pi/m_{ij}$ between hyperplanes

Example: The group $F_4$

Simple integer labels:

4
1 2 3
2 3 4
3 4 3

Custom labels:

4
vertices labels: t s1 s2 s3
t s1 3
s1 s2 4
s2 s3 3

Command-Line Usage

coxiter -i <input_file> [options]

Essential Options

Mandatory:

• -i <path>: Path to input file encoding the Coxeter graph

Computation flags:

• -full: Compute all invariants (Euler characteristic, f-vector, compactness, finite volume, growth series, growth rate)
• -arithmeticity or -a: Test arithmeticity (requires non-cocompact group)
• -compactness or -c: Test cocompactness only
• -fv: Test finite covolume only
• -g: Compute growth series only
• -growthrate: Compute growth rate (requires PARI library)
• -pgm: Print the Gram matrix

Output control:

• -o <basename>: Base name for output files (creates <basename>.output and optionally <basename>.jpg)
• -of: Write output to file (requires -o)
• -oformat <format>: Format for mathematical output (generic, gap, latex, mathematica, pari)

Advanced:

• -drop <vertex>: Work on subgraph by removing specified vertex (can be used multiple times)
• -drawgraph: Generate GraphViz visualization (requires -o, Unix only)
• -pcg: Print Coxeter graph structure
• -writegraph: Write graph in CoxIter format

Output Interpretation

Key Output Fields

Cocompact: yes/no

• Indicates whether the group acts cocompactly (fundamental domain compact)

Finite covolume: yes/no/?

• Indicates whether the group has finite covolume
• ? means unable to determine

f-vector: $(f_0, f_1, \ldots, f_n)$

• $f_k$ = number of $k$-dimensional faces of the fundamental polyhedron
• $f_0$ = vertices, $f_1$ = edges, ..., $f_n$ = $n$-dimensional faces

Euler characteristic: Rational number

• $\chi = \sum_{k=0}^n (-1)^k f_k$
• For even $n$: $\text{Covolume} = \pi^{n/2} \cdot \chi / \text{(normalization factor)}$

Growth series: Rational function $f(x) = P(x)/Q(x)$

• Coefficients give number of elements of each length
• Perron/Pisot/Salem classification of growth rate

Arithmetic: yes/no/?

• For non-cocompact groups, indicates arithmeticity
• ? often appears for graphs with dotted edges (requires manual verification)

Critical Mathematical Conventions

Gram Matrix

CoxIter computes the Gram matrix of the bilinear form. For hyperbolic Coxeter groups, this is:

• Symmetric matrix $(g_{ij})$ with $g_{ii} = 1$
• $g_{ij} = -\cos(\pi/m_{ij})$ for finite $m_{ij}$
• $g_{ij} = -1$ for $m_{ij} = \infty$ (parallel hyperplanes)
• $g_{ij} = -\cosh(d_{ij})$ for dotted edges (ultra-parallel, distance $d_{ij}$)

The Gram matrix signature determines the geometry:

• Signature $(n, 1)$: Hyperbolic space $H^n$
• Signature $(n, 0)$: Spherical (finite Coxeter group)
• Signature $(n-1, 1)$: Euclidean (affine Coxeter group)

Dotted Edges

A dotted edge (weight 1) represents ultra-parallel hyperplanes:

• Hyperplanes have a common perpendicular
• Distance encoded in Gram matrix entry: $g_{ij} = -\cosh(d_{ij})$
• For arithmeticity testing, dotted edges introduce variables that must satisfy algebraic conditions

Coxeter Graph Types

CoxIter identifies subgraphs to determine properties:

• Spherical graphs: Finite Coxeter groups (positive definite Gram matrix)
• Euclidean graphs: Affine Coxeter groups (positive semidefinite Gram matrix)
• Products of these determine compactness and finite volume

Common Use Cases

Compute All Invariants

coxiter -i graphs/mygroup.coxiter -full -o results/mygroup

Check Arithmeticity

coxiter -i graphs/mygroup.coxiter -a -pgm -o results/mygroup

Note: For graphs with dotted edges, arithmeticity may return ?. The output provides algebraic conditions that must be verified manually.

Compute Growth Rate

coxiter -i graphs/mygroup.coxiter -g -growthrate

Work on Subgraph

# Remove vertices 3 and 5, compute invariants for resulting subgraph
coxiter -i graphs/mygroup.coxiter -drop 3 -drop 5 -full

Export Gram Matrix

# In Mathematica format
coxiter -i graphs/mygroup.coxiter -pgm -oformat mathematica

Examples

Example 1: Bugaenko's group $B_8$

File 8-Bugaenko.coxiter:

11 8
1 2 5
2 3 3
3 4 3
4 5 3
5 6 3
6 7 3
7 8 3
8 9 5
4 10 3
6 11 3
10 11 1

Command:

coxiter -i graphs/8-Bugaenko.coxiter -full -o graphs/8-Bugaenko

Key outputs:

• Cocompact: yes
• Finite covolume: yes
• f-vector: (41, 164, 316, 374, 294, 156, 54, 11, 1)
• Euler characteristic: 24187/8709120000
• Growth rate: ~2.423 (Perron number)

Example 2: Arithmeticity Test with Dotted Edges

File with custom labels and dotted edges (weight 1):

7 3
vertices labels: 1 2 3 4 5 6 7
1 2 3
2 3 4
2 4 1
3 5 1
1 6 1
4 6 0
5 6 1
1 7 1
3 7 0

Command:

coxiter -i graphs/3-testArithmeticity.coxiter -a -pgm

Output includes:

• Gram matrix with variables for dotted edge weights
• Algebraic conditions for arithmeticity
• Manual verification required with specific values

Test Data

The CoxIter repository includes tests.txt containing data about a large set of hyperbolic Coxeter groups. CoxIter is run for each graph, and computed values are compared to expected values. This provides:

• Thousands of verified examples spanning various families (Bugaenko groups, Tumarkin pyramids, quadratic form unit groups, simplices)
• Expected Euler characteristic values, f-vectors, compactness classifications, and growth series
• Reference: https://github.com/rgugliel/CoxIter/blob/master/testing/tests.txt

References

• CoxIter: Guglielmetti, R. "CoxIter – Computing invariants of hyperbolic Coxeter groups", LMS Journal of Computation and Mathematics, 2015.
• Documentation: https://rgugliel.github.io/CoxIter/
• Source: https://github.com/rgugliel/CoxIter