How Groups Grow

Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory.

Preface; 1. Introduction; 2. Some group theory; 3. Groups of linear growth; 4. The growth of nilpotent groups; 5. The growth of soluble groups; 6. Linear groups; 7. Asymptotic cones; 8. Groups of polynomial growth; 9. Infinitely generated groups; 10. Intermediate growth: Grigorchuk's first group; 11. More groups of intermediate growth; 12. Growth and amenability; 13. Intermediate growth and residual finiteness; 14. Explicit calculations; 15. The generating function; 16. The growth of free products; 17. Conjugacy class growth; Research problems; References.

This book introduces the subject of the growth of groups from scratch, starting with basic definitions and culminating in the seminal results of Gromov and Grigorchuk and more. It is valuable reading for researchers from graduate students up who want to be acquainted with contemporary group theory.

Avinoam Mann is a Professor Emeritus at his Alma Mater, the Einstein Institute of Mathematics in the Hebrew University of Jerusalem. He has published over a hundred papers on group theory and co-authored the influential book Analytic Pro-p Groups.