# discgrp.5gv - Man Page

discrete group file formats

## Synopsis

An ascii file representation of a discrete geometric group.

## Summary

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

(group borromean )

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 } )

## Files

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).

## Authors

Charlie Gunn.

## Bugs

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