Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume / Roelof Bruggeman and Anke Dorothea Pohl.

Format:

Book

Author/Creator:

Bruggeman, Roelof W., 1944- author.

Pohl, Anke, author.

Series:

Memoirs of the American Mathematical Society ; Volume 287.

Memoirs of the American Mathematical Society : ; Volume 287

Language:

English

Subjects (All):

Number theory.

Number theory--Data processing.

Physical Description:

1 online resource (186 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"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"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

Motivational background

Aim of this monograph

Acknowledgement

Part 1. Preliminaries, properties of period functions, and some insights

Chapter 2. Notations

Chapter 3. Elements from hyperbolic geometry

3.1. Models and isometries

3.2. Classification of isometries

3.3. Cusps, funnels, limit set, and ordinary points

3.4. Geodesics, resonances, and the Selberg zeta function

3.5. Intervals and rounded neighborhoods

Chapter 4. Hecke triangle groups with infinite covolume

Chapter 5. Automorphic forms

5.1. Funnel forms of different types

5.2. Fourier expansion

Chapter 6. Principal series

6.1. Regularity at infinity

6.2. Presheaves and sheaves

6.3. Holomorphic extensions

Chapter 7. Transfer operators and period functions

7.1. Discretizations and transfer operators

7.2. Slow transfer operators

7.3. Period functions

7.4. Real and complex period functions

7.5. Fast transfer operators

7.6. One-sided averages

7.7. Convergence and meromorphic extension of fast transfer operators

7.8. Spaces of complex period functions

Chapter 8. An intuition and some insights

Part 2. Semi-analytic cohomology

Chapter 9. Abstract cohomology spaces

9.1. Standard group cohomology

9.2. Cohomology on an invariant set

9.3. Relation to parabolic cohomology spaces

Chapter 10. Modules

10.1. Modules of semi-analytic functions

10.2. Submodules of semi-analytic vectors

10.3. Conditions on cocycles

10.4. Cohomological interpretation of the singularity condition

Part 3. Automorphic forms and cohomology

Chapter 11. Invariant eigenfunctions via a group cohomology

Chapter 12. Tesselation cohomology

12.1. Choice of a tesselation, and cohomology

12.2. Relation to group cohomology

12.3. Mixed cohomology spaces.

Chapter 13. Extension of cocycles

Chapter 14. Surjectivity I: Boundary germs

14.1. Analytic boundary germs and semi-analytic modules

14.2. Cohomology classes attached to funnel forms

14.3. Representatives of boundary germs

Chapter 15. Surjectivity II: From cocycles to funnel forms

15.1. From a cocycle to an invariant eigenfunction

15.2. A cocycle on an orbit of ordinary points

15.3. Isomorphisms

Chapter 16. Relation between cohomology spaces

Chapter 17. Proof of Theorem D

From funnel forms to cocycle classes on the invariant set

From cocycle classes on to funnel forms

Proof of Theorem D

Part 4. Transfer operators and cohomology

Chapter 18. The map from functions to cocycles

Chapter 19. Real period functions and semi-analytic cocycles

Chapter 20. Complex period functions and semi-analytic cohomology

Chapter 21. Proof of Theorem E

Part 5. Proofs of Theorems A and B, and a recapitulation

Part 6. Parity

Chapter 22. The triangle group in the projective general linear group

22.1. Two actions of the projective general linear group

22.2. The triangle group

Chapter 23. Odd and even funnel forms, cocycles, and period functions

23.1. Odd and even funnel forms

23.2. Odd and even cocycles

23.3. Odd and even period functions

Chapter 24. Isomorphisms with parity

Part 7. Complements and outlook

Chapter 25. Fredholm determinant of the fast transfer operator

Chapter 26. Outlook

Bibliography

Index of terminology

List of notations

Back Cover.

Notes:

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Bruggeman, Roelof Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume

ISBN:

9781470475390

1470475391