Estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles

TL;DR

This paper establishes exponential decay estimates for the restriction of automorphic forms on hyperbolic manifolds to lower-dimensional hyperbolic geodesic cycles, expanding the forms into eigenfunctions of the Laplacian.

Contribution

It provides new exponential decay bounds for the coefficients in the eigenfunction expansion of automorphic forms restricted to hyperbolic geodesic cycles.

Findings

Exponential decay estimates for restriction coefficients

Expansion of automorphic forms into eigenfunctions on geodesic cycles

Application to hyperbolic manifolds and their cycles

Abstract

We find estimates for the restriction of automorphic forms on hyperbolic manifolds to compact geodesic cycles. The geodesic cycles we study are themselves hyperbolic manifolds of lower dimension. The restriction of an automorphic form to such a geodesic cycle can be expanded into eigenfunctions of the Laplacian on the geodesic cycle. We prove exponential decay for the coefficients in this expansion.