# The spectrum of a solenoid

## TL;DR

This paper investigates the spectral properties of the Laplacian on covering solenoids derived from sequences of finite coverings of Riemannian manifolds, establishing a key spectral equality and relating it to Selberg's conjecture.

## Contribution

It proves that the Laplacian spectrum on the covering solenoid equals the closure of the spectra of the sequence's manifolds, providing a new perspective on Selberg's conjecture.

## Key findings

- Spectrum of the Laplacian on the covering solenoid equals the closure of the union of spectra.
- Provides an equivalent formulation of Selberg's 1/4 conjecture.
- Establishes a link between geometric coverings and spectral theory.

## Abstract

Given a sequence of regular finite coverings of complete Riemannian manifolds, we consider the covering solenoid associated with the sequence. We study the leaf-wise Laplacian on the covering solenoid. The main result is that the spectrum of the Laplacian on the covering solenoid equals the closure of the union of the spectra of the manifolds in the sequence. We offer an equivalent statement of Selberg's 1/4 conjecture.