Eigenfunctions of transfer operators and automorphic forms for Hecke triangle groups of infinite covolume

TL;DR

This paper explores the deep connections between automorphic forms, cohomology, and transfer operator eigenfunctions for Hecke triangle groups of infinite covolume, revealing insights into their spectral and dynamical properties.

Contribution

It introduces cohomological interpretations for automorphic forms and establishes explicit isomorphisms linking these forms, cohomology, and transfer operator eigenfunctions for infinite covolume Hecke groups.

Findings

Explicit isomorphisms between automorphic forms and eigenfunctions

Cohomological interpretations for automorphic forms

Insights into spectral and geodesic flow relations

Abstract

We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.