Functions automorphic on large domains

Abstract

For a discontinuous group Γ ⊂ SL ( 2 , R ) \Gamma \subset {\text {SL}}(2,R) , Poincaré produced a corresponding nonconstant automorphic form, meromorphic on the open upper half plane Π + {\Pi ^ + } . When the domain of meromorphicity grows larger than Π + {\Pi ^ + } , the type of group which can support an automorphic form is restricted, and the corresponding forms are generally quite simple. A complete analysis of this phenomenon is presented, with examples which show results are best possible.