Subgroups of Teichmüller Modular Groups
About this Title
Nikolai V. Ivanov. Translated by E. J. Primrose
Publication: Translations of Mathematical Monographs
Publication Year: 1992; Volume 115
ISBNs: 978-0-8218-1968-5 (print); 978-1-4704-4526-3 (online)
MathSciNet review: MR1195787
MSC: Primary 57M50; Secondary 20F38, 30F60, 57N05
Teichmüller modular groups, also known as mapping class groups of surfaces, serve as a meeting ground for several branches of mathematics, including low-dimensional topology, the theory of Teichmüller spaces, group theory, and mathematical physics. The present work focuses mainly on the group-theoretic properties of these groups and their subgroups. The technical tools come from Thurston's theory of surfaces—his classification of surface diffeomorphisms and the theory of measured foliations on surfaces. The guiding principle of this investigation is a deep analogy between modular groups and linear groups. For some of the central results of the theory of linear groups (such as the theorems of Platonov, Tits, and Margulis-Soifer), the author provides analogous results for the case of subgroups of modular groups. The results also include a clear geometric picture of subgroups of modular groups and their action on Thurston's boundary of Teichmüller spaces. Aimed at research mathematicians and graduate students, this book is suitable as supplementary reading in advanced graduate courses.
Research mathematicians and graduate students.
Table of Contents
- Chapter 1. Diffeomorphisms acting trivially...
- Chapter 2. Preliminary information from the Theory of Surfaces
- Chapter 3. The action of pure diffeomorphisms on the Thurston Boundary. I
- Chapter 4. The action of pure diffeomorphisms on the Thurston Boundary. II
- Chapter 5. Pseudo-Anosov elements in irreducible subgroups of the group.....
- Chapter 6. Irreducible subgroups of the group....
- Chapter 7. The cutting of surfaces, and reduction systems
- Chapter 8. Free and Abelian subgroups
- Chapter 9. Maximal subgroups of infinite index
- Chapter 10. Frattini subgroups
- Chapter 11. Exercises