**Translations of Mathematical Monographs**

1992;
127 pp;
Softcover

MSC: Primary 57;
Secondary 20; 30; 32

**Print ISBN: 978-0-8218-1968-5
Product Code: MMONO/115.S**

List Price: $56.00

AMS Member Price: $44.80

MAA member Price: $50.40

# Subgroups of Teichmüller Modular Groups

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*Nikolai V. Ivanov*

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, more recently, 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 material in advanced graduate courses.

#### Readership

Research mathematicians and graduate students.

#### Reviews & Endorsements

Results … are collected here in a systematic study which is self-contained … The exposition is very clear and the proofs are given with full details.

-- Mathematical Reviews

This is a genuine research text, thoroughly planned and executed; the ideas are clearly outlined, background information and dependence on prior material are comprehensively listed, theorems are proved in appropriate detail, and illuminating comments are often placed at chapter's end comparing the present version with work of other authors … must be welcome as a slim volume of poetry on a favourite topic, and belongs on any surface topologist's shelf.

-- Bulletin of the London Mathematical Society

An impressive and well-written book on persistent work of the author over the last decade … both interesting for its results as well as for its methods and gives an excellent opportunity to see Thurston's theory of surface diffeomorphisms at work.

-- Zentralblatt MATH

Ivanov develops a rich and little-known theory, taking the newcomer to the start-point for research. To be sure the reader must work hard, but the proofs are complete and detailed.

-- Joan Birman, Columbia University