discrete group file formats

An ascii file representation of a discrete geometric group.

This file format applies to a wide variety of groups encountered in topology and
geometry. At the beginning of the file is the string DISCGRP, in conformity
with other OOGL file formats (see oogl(5)). The remainder of the file is a
series of parenthesized expressions.

The file format is based on an expandable set of key markers and values. A
typical entry would be

where parentheses enclose the expression. The string 'group' is called the key
marker. The remaining text contains a value, in this case a character string
which is the name of the group. Key markers currently recognized and their
values include:

group A name for this group.
comment An arbitrary comment [maximum length 256 bytes]
[must be enclosed in double quotes]
attribute One of a several keyword identifiers including:
euclidean is a euclidean group
hyperbolic is a hyperbolic group
spherical is a spherical group
transposed the matrices are transposed
finite this is a finite group
model In the case that 'hyperbolic' is an attribute, and the dimension is
three, this keyword sets the model that is
expected for reading subsequent isometries, i.e., expects elements of
SL(2,C) if the model is upperhalfspace or conformalball and O(3,1,R) if it is
projective.
projective default
upperhalfspace
conformalball [not currently implemented]
display These values set characteristics of the the display of the discrete
group.
centercam keeps the camera centered in central dirichlet domain.
zcull removes tiles which lie behind the camera.
showcam if centercam is set, displays camgeom, if provided (see below).
dimn The dimension of the space.
ngens Number of generators.
gens List of generators.
nels Number of group elements.
els List of group elements.
wafile Word acceptor file (in DEAGL format version 2.2)
camgeom An OOGL-format geometry description to represent the observer.
geom An OOGL-format geometry description. Will replace the default Dirichlet domain computation.

String-matching of key markers and attribute values is case-insensitive.

Currently all matrices must be 4x4 real matrices. Hence the only valid
dimensions are 2,3, or 4. The typical case is dimension 3 where there are
projective models for hyperbolic, euclidean, and spherical geometry. Dimension
2 groups should leave the z-coordinate unchanged.

Matrices are considered transposed if they are considered to act on column
vectors as in: M.v = v'. This is the usual mathematical convention convention.
So, if you use this convention you must mark your matrices as being
transposed.

Group elements are specified by an optional character string followed by a list
of 16 real numbers. For the case of generators, the character string
represents the name of that generator. Its main use is with the automatic
group software: the name used here should match that of the generator in the
word acceptor file. It should, for example, be a single character for the
software to work reliably. For composite elements, the string is the word in
the generators which describes this group element.

OOGL libraries expect discrete group files to use the '.dgp' suffix.

A file which describes the group of the three dimensional torus follows:

DISCGRP
(group 3torus )
(comment " This is a test of the new discrete group library. " )
(attribute Euclidean )
(display centercam )
(enumdepth 3 ) # look at all words of length 3 or less in generators
(enumdist 10.0 ) # and which don't move the origin more than 10.0
(dimn 3 )
(ngens 3 )
(gens
a
1 0 0 0
0 1 0 0
0 0 1 0
1 0 -0.1 1
b
1 0 0 0
0 1 0 0
0 0 1 0
0.1 1 0 1
c
1 0 0 0
0 1 0 0
0 0 1 0
0 0.2 1 1
)
(cpoint 0.000000 0.000000 0.000000 1.000000 )
(camgeom
{ = OFF
5 2 5
0 0 0
-0.1 0 0.5
0.1 0 0.5
0 -0.1 0.5
0 0.1 0.5
3 0 1 2 200 200 0 .8
3 0 3 4 0 200 200 .8
}
)

Example group files are to be found in the OOGL release in ${GEOM}/data/groups,
where e.g., GEOM = /u/gcg/ngrap. See also discgrp(3) and automata(1).

Charlie Gunn.

For dimension 2, the code should read and write 3 x 3 matrices.