A Spanning Set for the space of Super Cusp forms

TL;DR

This paper constructs a spanning set for super cusp forms on a complex symmetric super domain of rank 1, using advanced dynamical systems techniques and Fourier analysis, with elements corresponding to closed geodesics.

Contribution

It introduces a novel method to generate a spanning set for super cusp forms leveraging a generalized Anosov closing lemma and Fourier decomposition.

Findings

Spanning set elements correspond to closed geodesics.

Number of spanning elements grows linearly with geodesic length.

Generalization of the Anosov closing lemma for super domains.

Abstract

Aim of this article is the construction of a spanning set for the space of super cusp forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. The main ingredients are a generalization of the Anosov closing lemma for partially hyperbolic diffeomorphisms and an unbounded realization of B, in particular Fourier decomposition at the cusps mapped to infinity via a partial Cayley transformation. The elements of the spanning set are in finite-to-one correspondence with closed geodesics, the number of elements corresponding to a geodesic growing linearly with its length.