Cusp forms associated to loxodromic elements of Kleinian groups
Let \(\Gamma\) be a finitely generated nonelementary Kleinian group with region of discontinuity \(\Omega\). Suppose that \(q\geq 2\) is an integer and \(A\in \Gamma\) is loxodromic with attracting fixed point \(\alpha\) and repulsive fixed point \(\beta\). Let \[ g_{\alpha,\beta}(z):=\frac{\alpha -\beta}{(z-\alpha)(z-\beta)} \] and form the relative Poincaré series associated with A, \[ \phi_ A(z)=\sum_{\gamma \in \setminus \Gamma}(g_{\alpha,\beta}(\gamma z)\gamma '(z))^ q\quad (z\in \Omega). \] Then \(\phi_ A\in {\mathbb{A}}_ q(\Omega,\Gamma)\), the space of cusp forms for the group \(\Gamma\) of weight -2q. The paper under review gives very satisfactory results on spanning sets and vanishing of linear combinations, of the series \(\phi_ A\), namely: Theorem 1. The functions \(\phi_ A\) (A\(\in \Gamma\), A loxodromic) span the space \({\mathbb{A}}_ q(\Omega,\Gamma)\). (In fact, this is just a special case of a more general result for arbitrary nonelementary Kleinian groups stated on p. 604.) Theorem 2. Suppose in addition that we are given a finite presentation for \(\Gamma\) and that a certain map introduced by Bers is surjective. Then the identical vanishing of an arbitrary linear combination \[ \phi =\sum^{n}_{j=1}\lambda_ j\phi_{A_ j} \] (\(\lambda\) \({}_ 1,...,\lambda_ n\in {\mathbb{C}}\), \(A_ 1,...,A_ n\in \Gamma\) loxodromic) can be decided by a (finite) algorithm involving linear algebra and the presentation of \(\Gamma\). (The precise meaning of Theorem 2 is carefully explained in the paper.) There are several interesting related results that cannot be given here. - Similar problems for Poincaré series rather than relative Poincaré series were discussed earlier by the author [Acta Math. 153, 47-116 (1984; Zbl 0543.30037)].
scientific article; zbMATH DE number 3941898