Eigenfunctions of transfer operators and cohomology

TL;DR

This paper explores the relationship between eigenfunctions of transfer operators and cohomology groups for the modular group, revealing a bijective correspondence linked to continued fraction algorithms.

Contribution

It establishes a novel connection between transfer operator eigenfunctions and cohomology groups, advancing understanding of their spectral and algebraic properties.

Findings

Eigenfunctions with eigenvalues ±1 correspond to cohomology classes.

A bijective correspondence is shown between transfer operator eigenfunctions and cohomology groups.

The work relates spectral properties of transfer operators to modular group cohomology.

Abstract

The eigenfunctions with eigenvalues 1 or -1 of the transfer operator of Mayer are in bijective correspondence with the eigenfunctions with eigenvalue 1 of a transfer operator connected to the nearest integer continued fraction algorithm. This is shown by relating these eigenspaces of these operators to cohomology groups for the modular group with coefficients in certain principal series representations.