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Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume
About this Title
Roelof Bruggeman and Anke Dorothea Pohl
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 287, Number 1423
ISBNs: 978-1-4704-6545-2 (print); 978-1-4704-7539-0 (online)
DOI: https://doi.org/10.1090/memo/1423
Published electronically: June 28, 2023
Keywords: automorphic forms,
funnel forms,
transfer operator,
cohomology,
mixed cohomology,
period functions,
Hecke triangle groups,
infinite covolume
Table of Contents
Chapters
- 1. Introduction
1. Preliminaries, properties of period functions, and some insights
- 2. Notations
- 3. Elements from hyperbolic geometry
- 4. Hecke triangle groups with infinite covolume
- 5. Automorphic forms
- 6. Principal series
- 7. Transfer operators and period functions
- 8. An intuition and some insights
2. Semi-analytic cohomology
- 9. Abstract cohomology spaces
- 10. Modules
3. Automorphic forms and cohomology
- 11. Invariant eigenfunctions via a group cohomology
- 12. Tesselation cohomology
- 13. Extension of cocycles
- 14. Surjectivity I: Boundary germs
- 15. Surjectivity II: From cocycles to funnel forms
- 16. Relation between cohomology spaces
- 17. Proof of Theorem D
4. Transfer operators and cohomology
- 18. The map from functions to cocycles
- 19. Real period functions and semi-analytic cocycles
- 20. Complex period functions and semi-analytic cohomology
- 21. Proof of Theorem E
5. Proofs of Theorems A and B, and a recapitulation
6. Parity
- 22. The triangle group in the projective general linear group
- 23. Odd and even funnel forms, cocycles, and period functions
- 24. Isomorphisms with parity
7. Complements and outlook
- 25. Fredholm determinant of the fast transfer operator
- 26. Outlook
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.- Alexander Adam and Anke Pohl, A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions, Ergodic Theory Dynam. Systems 40 (2020), no. 3, 612–662. MR 4059792, DOI 10.1017/etds.2018.51
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