Given a pregroup \(P\) there is a universal group \(\mathcal{U}(P)\) that contains \(P\). The concept of a pregroup presentation is a generalisation of presentations over the free group, that is a pregroup presentation is a way of defining a group as a quotient of a universal group over a pregroup by giving relator words over the pregroup.
For the purposes of the RSym tester we introduce some more concepts.
A location on a pregroup relator \( w = a_1a_2\ldots a_n\) is an index \(i\) between \(1\) and \(n\) and denotes the location between \(a_i\) (the InLetter (3.2-2)) and \(a_{i+1}\) (the OutLetter (3.2-3)), where the relator is considered cyclically, that is, when \(i=n\) then the outletter is \(a_1\).
A place \(R(L, x, C)\) on a pregroup relator \(R\) is a location (3.1-1) together with a letter from the pregroup and a colour, which is either red or green.
‣ IsPregroupLocation( arg ) | ( filter ) |
Returns: true or false
‣ InLetter( arg ) | ( attribute ) |
‣ OutLetter( arg ) | ( attribute ) |
‣ Places( arg ) | ( attribute ) |
‣ NextLocation( arg ) | ( attribute ) |
‣ PrevLocation( arg ) | ( attribute ) |
‣ __ID( arg ) | ( attribute ) |
‣ NewPregroupPresentation( pregroup, relators ) | ( function ) |
Returns: a pregroup presentation
Creates a pregroup presentation over the pregroup with relators relators.
‣ PregroupPresentationFromFp( F, rred, rgreen ) | ( function ) |
Returns: a pregroup presentation
Creates a pregroup presentation over the pregroup defined by F and rred with relators rgreen.
‣ PregroupPresentationToFpGroup( presentation ) | ( function ) |
Returns: a finitely presented group
Converts the pregroup presentation presentation into a finitely presented group.
‣ IsPregroupPresentation( arg ) | ( filter ) |
Returns: true or false
‣ ( arg ) | ( filter ) |
Returns: true or false
‣ RSymTestOp( presentation, epsilon ) | ( operation ) |
Test the group presented by presentation for hyperbolicity using the RSym tester with parameter epsilon.
‣ RSymTest( args... ) | ( function ) |
This is a wrapper for RSymTestOp (3.5-1). If the first argument given is a free group, the second and third lists of words over the free group, and the fourth a rational, then this function creates a pregroup presentation from the input data and invokes RSymTestOp (3.5-1) on it. If the first argument is a pregroup presentation and the second argument is rational number, then it invokes RSymTestOp (3.5-1) on that input.
‣ IsHyperbolic( presentation ) | ( operation ) |
‣ IsHyperbolic( presentation, epsilon ) | ( operation ) |
‣ IsHyperbolic( F, rred, rgreen, epsilon ) | ( operation ) |
Tests a given presentation for hyperbolicity using the RSym test procedure.
‣ PregroupPresentationToKBMAG( presentation ) | ( function ) |
Returns: A KBMAG rewriting system
Turns the pregroup presentation presentation into valid input for Knuth-Bendix rewriting using KBMAG. Only available if the kbmag package is available.
‣ PregroupPresentationToStream( stream, presentation ) | ( function ) |
Writes the pregroup presentation presentation to stream.
gap> T := TriangleGroup(2,3,7);; gap> str := "";; stream := OutputTextString(str, true);; gap> PregroupPresentationToStream(stream, T); gap> Print(str); rec( rels := [ [ 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3 ] ], table := [ [ 1, 2, 3, 4 ], [ 2, 1, 0, 0 ], [ 3, 0, 4, 1 ], [ 4, 0, 1, 3 ] ] );
‣ PregroupPresentationFromStream( stream ) | ( function ) |
Returns: A pregroup presentation
Reads a pregroup presentation from an input stream in the same format that PregroupPresentationToStream (3.6-2) uses.
gap> stream := InputTextString(str); InputTextString(0,146) gap> PregroupPresentationFromStream(stream); <pregroup presentation with 3 generators and 1 relators>
‣ PregroupPresentationToSimpleStream( stream, presentation ) | ( function ) |
Writes the pregroup presentation presentation to stream. Uses a simpler format than PregroupPresentationToStream (3.6-2)
‣ PregroupPresentationToFile( filename, presentation ) | ( function ) |
Writes the pregroup presentation presentation to file with name filename.
‣ PregroupPresentationFromFile( filename ) | ( function ) |
Reads a pregroup presentation from file with filename.
‣ PregroupPresentationToSimpleFile( stream, presentation ) | ( function ) |
Writes the pregroup presentation presentation to file with name filename in a simple format.
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