Asymptotics of the triangle groups and the associated spectra

Abstract

We consider the asymptotics of the $(p,q,r)$-triangle groups as one of the vertices goes to a cusp $ptoinfty$. We introduce new coordinates on the fundamental domain of the $(p,q,r)$-triangle group which in the limit approach the natural coordinates on the $(infty, q,r)$-triangle group and show that in these coordinates the fundamental solutions on the $(p,q,r)$-triangle group approach the fundamental solutions on the $(infty, q,r)$-triangle group. We use these coordinates to study how eigensolutions in the co-compact case accumulate onto the continuous spectrum in the non co-compact case.