Selberg Formula for Cofinite Groups and the Roelke Conjecture

TL;DR

This paper corrects previous errors and proves the Roelke Conjecture for cofinite groups by establishing a formula linking the discrete spectrum and resonance spectrum of the Laplace operator.

Contribution

It introduces a new formula connecting spectra for cofinite groups and provides a corrected proof of the Roelke Conjecture.

Findings

Established a formula linking discrete and resonance spectra

Provided a corrected proof of the Roelke Conjecture

Clarified the spectral relationships for cofinite groups

Abstract

As the reviewer have pointed out, the proof of Roelke Conjecture contains an error. For cofinite groups, we obtain a formula connecting the discrete spectrum of Laplace operator and the resonance spectrum. Using this formula, we give a proof of the Roelke conjecture. In this version, we make several corrections in Theorem 1 and Lemma 5.