The Cremona Group and Its Subgroups

ISBN-13: 9781470460129

Julie Deserti

Opis

The goal of this book is to present a portrait of the $n$-dimensional Cremona group with an emphasis on the 2-dimensional case. After recalling some crucial tools, the book describes a naturally defined infinite dimensional hyperbolic space on which the Cremona group acts. This space plays a fundamental role in the study of Cremona groups, as it allows one to apply tools from geometric group theory to explore properties of the subgroups of the Cremona group as well as the degree growth and dynamical behavior of birational transformations. The book describes natural topologies on the Cremona group, codifies the notion of algebraic subgroups of the Cremona groups and finishes with a chapter on the dynamics of their actions. This book is aimed at graduate students and researchers in algebraic geometry who are interested in birational geometry and its interactions with geometric group theory and dynamical systems.

Spis treści

• Introduction
• An isometric action of the Cremona group on an infinite dimensional hyperbolic space
• Algebraic subgroups of the Cremona group
• Generators and relations of the Cremona group
• Algebraic properties of the Cremona group
• Finite subgroups of the Cremona group
• Uncountable subgroups of the Cremona group
• Consequences of the action of the Cremona group of an infinite dimensional hyperbolic space
• Big subgroups of autmorphisms ``of positive entropy''
• Bibliography
• Index
• Index notations