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6 Reidemeister zeta functions

6.1 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.

Please note that the functions below are only implemented for endomorphisms of finite groups.

6.1-1 ReidemeisterZetaCoefficients

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.

6.1-2 IsRationalReidemeisterZeta

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

6.1-3 ReidemeisterZeta

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

6.1-4 PrintReidemeisterZeta

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

This is often more readable than evaluating ReidemeisterZeta in an indeterminate, and does not require rationality.

gap> khi := GroupHomomorphismByImages( G, G, [ (1,2,3,4,5), (4,5,6) ],
>  [ (1,2,6,3,5), (1,4,5) ] );;
gap> ReidemeisterZetaCoefficients( khi );
[ [ 7 ], [  ] ]
gap> IsRationalReidemeisterZeta( khi );
true
gap> ReidemeisterZeta( khi );
function( s ) ... end
gap> s := Indeterminate( Rationals, "s" );;
gap> ReidemeisterZeta( khi )(s);
(1)/(-s^7+7*s^6-21*s^5+35*s^4-35*s^3+21*s^2-7*s+1)
gap> PrintReidemeisterZeta( khi );
"(1-s)^(-7)"

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