Topics in Geometric Group Theory - de la Harpe, Pierre
- Collection: Lectures In Mathematics
- Format: Broché Voir le descriptif
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Présentation Topics In Geometric Group Theory Format Broché
- Livre Mathématiques
Résumé :
Groups as abstract structures were first recognized by mathematicians in the nineteenth century. Groups are, of course, sets given with appropriate "multiplications," and they are often given together with actions on interesting geometric objects. But groups are also interesting geometric objects by themselves. More precisely, a finitely-generated group can be seen as a metric space, the distance between two points being defined "up to quasi-isometry" by some "word length," and this gives rise to a very fruitful approach to group theory. In this book, Pierre de la Harpe provides a concise and engaging introduction to this approach, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe uses a hands-on presentation style, illustrating key concepts of geometric group theory with numerous concrete examples. The first five chapters present basic combinatorial and geometric group theory in a unique way, with an emphasis on finitely-generated versus finitely-presented/groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group," an infinite finitely-generated torsion group of intermediate growth which is becoming more and more important in group theory. Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students. problems range from slightly more difficult exercises to open research questions in the field. An extensive list of references directs readers to more advanced results as well as connections with other subjects.
Biographie:
Pierre de la Harpe is professor of mathematics at the Université de Genève, Switzerland. He is the author, coauthor, or coeditor of several books, including La propriété (T) de Kazhdan pour les groupes localement compacts and Sur les groupes hyperboliques d'après Mikhael Gromov.
Sommaire:
["Gauss'circle problem and Polya's random walks on lattices","Free products and free groups","Finitely-generated groups","Finitely-generated groups viewed as metric spaces","Finitely-presented groups","Growth of finitely-generated groups","Groups of exponential or polynomial growth","The first Grigorchuk group."]
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