The transfer operator for
and trivial character
possesses a finite group of symmetries generated by permutation matrices
with
.
Every such symmetry leads to a factorization of the Selberg zeta
function in terms of Fredholm determinants of a reduced transfer
operator. These symmetries are related to the group of automorphisms in
of the Maass wave
forms of
. For
the group
and
Selberg’s character
there exists just one nontrivial symmetry operator
. The
eigenfunctions of the corresponding reduced transfer operator with eigenvalue
are related to Maass forms that are even or odd, respectively, under a
corresponding automorphism. It then follows from a result of Sarnak and
Phillips that the zeros of the Selberg function determined by the eigenvalue
of the reduced
transfer operator stay on the critical line under deformation of the character. From numerical
results we expect that, on the other hand, all the zeros corresponding to the eigenvalue
are off this line for a
nontrivial character
.
Keywords
transfer operator, Hecke congruence subgroups, Maass wave
forms, character deformation, factorization of the Selberg
zeta function