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By a transfer operator approach to Maass cusp forms and the Selberg zetafunction for cofinite Hecke triangle groups, M. M\"oller and the author found afactorization of the Selberg zeta function into a product of Fredholmdeterminants of transfer-operator-like families: $Z(s) = \det(1-\mcL_s^+)\det(1-\mc L_s^-)$. In this article we show that the operator families$\mc L_s^\pm$ arise as families of transfer operators for the triangle groupsunderlying 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 onlyif there exists an even (resp. odd) Maass cusp form with eigenvalue $s(1-s)$.For nonarithmetic Hecke triangle groups, this result provides a new formulationof the Phillips-Sarnak conjecture on nonexistence of even Maass cusp forms.

Odd and even Maass cusp forms for Hecke triangle groups, and the billiard flow