chap3.txt

  
  3 Farey symbols and their properties
  
  A Farey symbol is a compact and useful way to represent a subgroup of finite
  index  in  SL_2(ℤ) from which one can deduce independent generators for this
  subgroup.  It  consists  of  two  components, namely a so-called generalised
  Farey  sequence  (gfs)  and  an  ordered  list  of labels, giving additional
  structure to the gfs.
  
  A  generalised  Farey  sequence  (g.F.S.)  is  an  ordered  list of the form
  -infinity, x_0, x_1, ... , x_n, infinity, where
  
  1.  the  x_i  =  a_i/b_i  are  rational  numbers in reduced form arranged in
  increasing order for i = 0, ... , n;
  
  2. x_0, ... , x_n ∈ Z, and some x_i = 0;
  
  3. we define x_-1=-infinity=-1/0 and x_n+1=infinity=1/0;
  
  4. a_i+1b_i-a_ib_i+1=1 for i=-1, ... ,n.
  
  The  ordered  list of labels of a Farey symbol gives an additional structure
  to  the gfs. The labels correspond to each consecutive pair of x_i's and are
  of the following types:
  
  1. even,
  
  2. odd,
  
  3. a natural number, which occurs in the list of labels exactly twice or not
  at all.
  
  Note that the actual values of numerical labels are not important; it is the
  pairing of two intervals that matters.
  
  The  package Congruence provides functions to construct Farey symbols by the
  given  generalised  Farey  sequence  and  corresponding  list of labels. The
  returned  Farey  symbol  will  belong to the category IsFareySymbol and will
  have the representation IsFareySymbolDefaultRep.
  
  
  3.1 Construction of Farey symbols
  
  3.1-1 FareySymbolByData
  
  FareySymbolByData( gfs, labels )  function
  
  This  constructor  creates the Farey symbol with the given generalized Farey
  sequence  and  list of labels. It also checks conditions from the definition
  of  Farey  symbol  and  returns an error if they are not satisfied. The data
  used   to   create   the   Farey   symbol   are  stored  as  its  attributes
  GeneralizedFareySequence (3.2-1) and LabelsOfFareySymbol (3.2-4).
  
    Example  
    
    gap> fs:=FareySymbolByData([infinity,0,1,2,infinity],[1,2,2,1]);                         
    [ infinity, 0, 1, 2, infinity ]
    [ 1, 2, 2, 1 ]
    
  
  
  3.1-2 IsValidFareySymbol
  
  IsValidFareySymbol( fs )  function
  
  This function is used in FareySymbolByData (3.1-1) to validate its output.
  
    Example  
    
    gap> IsValidFareySymbol(fs);
    true
    
  
  
  
  3.2 Properties of Farey symbols
  
  3.2-1 GeneralizedFareySequence
  
  GeneralizedFareySequence( fs )  attribute
  
  Returns the generalized Farey sequence gfs of the Farey symbol.
  
    Example  
    
    gap> GeneralizedFareySequence(fs);
    [ infinity, 0, 1, 2, infinity ]
    
  
  
  3.2-2 NumeratorOfGFSElement
  
  NumeratorOfGFSElement( gfs, i )  function
  Returns:  integer
  
  Returns  the  numerator  of  the i-th term of the generalised Farey sequence
  gfs:  for the 1st infinite entry returns -1, for the last one returns 1, for
  all other entries returns the usual numerator.
  
    Example  
    
    gap> List([1..5], i -> NumeratorOfGFSElement(GeneralizedFareySequence(fs),i));
    [ -1, 0, 1, 2, 1 ]
    
  
  
  3.2-3 DenominatorOfGFSElement
  
  DenominatorOfGFSElement( gfs, i )  function
  Returns:  integer
  
  Returns  the  denominator of the i-th term of the generalised Farey sequence
  gfs:  for  both  infinite  entries returns 0, for the other ones returns the
  usual denominator.
  
    Example  
    
    gap> List([1..5], i -> DenominatorOfGFSElement(GeneralizedFareySequence(fs),i));         
    [ 0, 1, 1, 1, 0 ]
    
  
  
  3.2-4 LabelsOfFareySymbol
  
  LabelsOfFareySymbol( fs )  attribute
  
  Returns  the list of labels of the Farey symbol. This list has "odd", "even"
  and paired integers as entries.
  
    Example  
    
    gap> LabelsOfFareySymbol(fs);
    [ 1, 2, 2, 1 ]