  By a transfer operator approach to Maass cusp forms and the Selberg zeta
function for cofinite Hecke triangle groups, M. M\"oller and the author found a
factorization of the Selberg zeta function into a product of Fredholm
determinants of transfer-operator-like families: $Z(s) = \det(1-\mc
L_s^+)\det(1-\mc L_s^-)$. In this article we show that the operator families
$\mc L_s^\pm$ arise as families of transfer operators for the triangle groups
underlying the Hecke triangle groups, and that for $s\in\C$, $\Rea s=\tfrac12$,
the operator $\mc L_s^+$ (resp. $\mc L_s^-$) has a 1-eigenfunction if and only
if there exists an even (resp. odd) Maass cusp form with eigenvalue $s(1-s)$.
For nonarithmetic Hecke triangle groups, this result provides a new formulation
of the Phillips-Sarnak conjecture on nonexistence of even Maass cusp forms.
