Let \(\Gamma\) be a finitely generated Fuchsian group of the first kind acting on the upper half-plane \({\mathfrak H}\), and let \(\Delta\) denote the Laplace-Beltrami operator for the hyperbolic metric acting on its natural domain in \(L^ 2(\Gamma \setminus {\mathfrak H})\). Suppose that \(\Gamma\) \(\setminus {\mathfrak H}\) is not compact. Then the so-called Roelcke-Selberg conjecture asserts that there exist infinitely many linearly independent cusp eigenfunctions of \(\Delta\). This conjecture is known to be true in many special cases, but for generic \(\Gamma\) it is still an open problem.
In the paper under review the authors consider the behaviour of a given cusp form u under a quasiconformal deformation of the group \(\Gamma\) and they investigate the problem whether u may be destroyed under such a real analytic deformation. The main result is Theorem 2.1 which gives a sufficient condition which ensures that for all sufficiently small \(t>0\) the deformed group \(\Gamma_ t\) has no cusp form whose eigenvalue is near the eigenvalue \(\lambda_ 0\) for u. The deformations considered here are defined by means of Beltrami differentials associated with weight 4 holomorphic cusp forms Q for \(\Gamma\). In fact, the authors even prove that except for a denumerable set of values of t, no new cusp forms are created by such deformations.
The sufficient condition of Theorem 2.1 is reformulated in Theorem 3.1 in terms of the Rankin-Selberg convolution of Q and u. The authors also give some examples and numerical calculations which indicate that for generic \(\Gamma\) the Roelcke-Selberg conjecture may be false.