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6 Reidemeister zeta and generating functions

Please note that the functions in this chapter are only implemented for endomorphisms of finite groups.

6.1 Decomposing Reidemeister number sequences

6.1-1 IteratedReidemeisterNumberDecomposition

Returns: two lists of integers.

For a finite group, the sequence of Reidemeister numbers of the iterates of endo1 and endo2, i.e. the sequence R(endo1,endo2), R(endo1^2,endo2^2), ..., is eventually periodic. Thus there exist a periodic sequence \((P_n)_{n \in \mathbb{N}}\) and an eventually zero sequence \((Q_n)_{n \in \mathbb{N}}\) such that

\[\forall n \in \mathbb{N}: R(\varphi^n,\psi^n) = P_n + Q_n.\]

This function returns two lists: the first list contains one period of the sequence \((P_n)_{n \in \mathbb{N}}\), the second list contains \((Q_n)_{n \in \mathbb{N}}\) up to the part where it becomes the constant zero sequence.

gap> G := PcGroupCode( 3913, 12 );;
gap> phi := GroupHomomorphismByImages( G, G, [ G.1, G.3 ], [ One(G), One(G) ] );;
gap> psi := GroupHomomorphismByImages( G, G, [ G.1, G.3 ], [ G.2, One(G) ] );;
gap> IteratedReidemeisterNumberDecomposition( phi );
[ [ 1 ], [  ] ]
gap> IteratedReidemeisterNumberDecomposition( phi, psi );
[ [ 12 ], [ -6 ] ]

6.2 Reidemeister zeta functions

Let \(\varphi,\psi\colon G \to G\) be endomorphisms such that \(R(\varphi^n,\psi^n) < \infty\) for all \(n \in \mathbb{N}\). Then the Reidemeister zeta function \(Z_{\varphi,\psi}(s)\) of the pair \((\varphi,\psi)\) is defined as

\[Z_{\varphi,\psi}(s) := \exp \sum_{n=1}^\infty \frac{R(\varphi^n,\psi^n)}{n} s^n.\]

6.2-1 IsRationalReidemeisterZetaFunction

Returns: true if the Reidemeister zeta function of endo1 and endo2 is rational, otherwise false.

6.2-2 ReidemeisterZetaFunction

Returns: the Reidemeister zeta function of endo1 and endo2 if it is rational, otherwise fail.

6.2-3 PrintReidemeisterZetaFunction

Returns: a string describing the Reidemeister zeta function of endo1 and endo2.

This is often more readable than evaluating ReidemeisterZetaFunction (6.2-2) in an indeterminate, and does not require rationality.

gap> IsRationalReidemeisterZetaFunction( phi );
true
gap> IsRationalReidemeisterZetaFunction( phi, psi );
false
gap> s := Indeterminate( Rationals, "s" );;
gap> ReidemeisterZetaFunction( phi )(s);
(1)/(-s+1)
gap> PrintReidemeisterZetaFunction( phi, psi );
"exp(-6*s)*(1-s)^(-12)"

6.3 Reidemeister generating functions

Let \(\varphi,\psi\colon G \to G\) be endomorphisms such that \(R(\varphi^n,\psi^n) < \infty\) for all \(n \in \mathbb{N}\). Then the Reidemeister generating function \(Z^*_{\varphi,\psi}(s)\) of the pair \((\varphi,\psi)\) is defined as

\[Z^*_{\varphi,\psi}(s) := \sum_{n=1}^\infty R(\varphi^n,\psi^n) s^n.\]

6.3-1 IsRationalReidemeisterGeneratingFunction

Returns: true if the Reidemeister generating function of endo1 and endo2 is rational, otherwise false.

6.3-2 ReidemeisterGeneratingFunction

Returns: the Reidemeister generating function of endo1 and endo2 if it is rational, otherwise fail.

6.3-3 PrintReidemeisterGeneratingFunction

Returns: a string describing the Reidemeister generating function of endo1 and endo2.

This is often more readable than evaluating ReidemeisterGeneratingFunction (6.3-2) in an indeterminate, and does not require rationality.

gap> IsRationalReidemeisterGeneratingFunction( phi, psi );
true
gap> ReidemeisterGeneratingFunction( phi, psi )( 2 );
-36
gap> PrintReidemeisterGeneratingFunction( phi, psi );
"(6*s^2+6*s)/(-s+1)"

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