Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume

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 operato...

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Main Authors: Bruggeman, Roelof, Pohl, Anke Dorothea
Format: eBook Book
Language:English
Published: Providence, Rhode Island American Mathematical Society 2023
Edition:1
Series:Memoirs of the American Mathematical Society
Subjects:
Number theory
Number theory-Data processing
infinite covolume
transfer operator
period functions
mixed cohomology
cohomology
automorphic forms
Hecke triangle groups
funnel forms
ISBN:9781470465452, 1470465450
ISSN:0065-9266, 1947-6221
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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.
AbstractList 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.
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Author Bruggeman, Roelof
Pohl, Anke Dorothea
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Copyright Copyright 2023 American Mathematical Society
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Keywords infinite covolume
transfer operator
period functions
mixed cohomology
cohomology
automorphic forms
Hecke triangle groups
funnel forms
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Notes Includes bibliographical references (p. 165-167) and index
July 2023, volume 287, number 1423 (first of 6 numbers)
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Snippet We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these...
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Number theory-Data processing
TableOfContents Introduction -- Preliminaries, properties of period functions, and some insights -- Notations -- Elements from hyperbolic geometry -- Hecke triangle groups with infinite covolume -- Automorphic forms -- Principal series -- Transfer operators and period functions -- An intuition and some insights -- Semi-analytic cohomology -- Abstract cohomology spaces -- Modules -- Automorphic forms and cohomology -- Invariant eigenfunctions via a group cohomology -- Tesselation cohomology -- Extension of cocycles -- Surjectivity I: Boundary germs -- Surjectivity II: From cocycles to funnel forms -- Relation between cohomology spaces -- Proof of Theorem D -- Transfer operators and cohomology -- The map from functions to cocycles -- Real period functions and semi-analytic cocycles -- Complex period functions and semi-analytic cohomology -- Proof of Theorem E -- Proofs of Theorems A and B, and a recapitulation -- Parity -- The triangle group in the projective general linear group -- Odd and even funnel forms, cocycles, and period functions -- Isomorphisms with parity -- Complements and outlook -- Fredholm determinant of the fast transfer operator -- Outlook
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
Title Eigenfunctions of Transfer Operators and Automorphic Forms for Hecke Triangle Groups of Infinite Covolume
URI https://www.ams.org/memo/1423/
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