Cusp forms associated to loxodromic elements of Kleinian groups
type: Item
instance of: scholarly article
title: Cusp forms associated to loxodromic elements of Kleinian groups
zbMATH Open document ID: 0587.30044
DOI: 10.1215/S0012-7094-85-05232-9
author: Irwin Kra
published in: Duke Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 30F35
zbMATH DE Number: 3941898
zbMATH Keywords: Eichler cohomology group
zbMATH Keywords: Bers spaces
zbMATH Keywords: Kleinian group
zbMATH Keywords: Poincaré series
MaRDI profile type: Publication
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full work available at URL: s0012-7094-85-05232-9
OpenAlex ID: W1520328879
description: scientific article; zbMATH DE number 3941898