# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1705.05612