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We characterize Maass cusp forms for any cofinite Hecke triangle group as 1-eigenfunctions of appropriate regularity of a transfer operator family. This transfer operator family is associated to a certain symbolic dynamics for the geodesic flow on the orbifold arising as the orbit space of the action of the Hecke triangle group on the hyperbolic plane. Moreover we show that the Selberg zeta function is the Fredholm determinant of the transfer operator family associated to an acceleration of this symbolic dynamics.

Populus nigra L. is a pioneer tree species of riparian ecosystems that is threatened with extinction because of the loss of its natural habitat. To evaluate the existing genetic diversity of P. nigra within ex-situ collections, we analyzed 675 P. nigra L. accessions from nine European gene banks with three amplified fragment length polymorphism (AFLP) and five microsatellite [or simple sequence repeat (SSR)] primer combinations, and 11 isozyme systems. With isozyme analysis, hybrids could be detected, and only 3% were found in the gene bank collection. AFLP and SSR analyses revealed effectively that 26% of the accessions were duplicated and that the level of clonal duplication varied from 0% in the French gene bank collection up to 78% in the Belgian gene bank collection. SSR analysis was preferred because AFLP was technically more demanding and more prone to scoring errors. To assess the genetic diversity, we grouped material from the gene banks according to topography of the location from which the accessions were originally collected (river system or regions separated by mountains). Genetic diversity was expressed in terms of the following parameters: percentage of polymorphic loci, observed and effective number of alleles, and Nei's expected heterozygosity or gene diversity (for AFLP). Genetic diversity varied from region to region and depended, to some extent, on the marker system used. The most unique alleles were identified in the Danube region (Austria), the Rhône region (France), Italy, the Rijn region (The Netherlands), and the Ebro region (Spain). In general, the diversity was largest in the material collected from the regions in Southern Europe. Dendrograms and principal component analysis resulted in a clustering according to topography. Material from the same river systems, but from different countries, clustered together. The genetic differentiation among the regions (F(st)/G(st)) was moderate.

Abstract. Let F be a square integrable Maaß form on the Siegel upper half space H of rank 2 for the Siegel modular group Sp 4 (Z) with Laplace eigenvalue λ. If, in addition, F is a joint eigenfunction of the Hecke algebra and Ω is a compact set in Sp 4 (Z)\H, we show the bound F | Ω ∞ ≪ Ω (1 + λ) 1−δ for some global constant δ > 0. As an auxiliary result of independent interest we prove new uniform bounds for spherical functions on semisimple Lie groups.

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