# Geometric Group Theory

###### Cornelia Druţu and Michael Kapovich
Publisher:
American Mathematical Society
Publication Date:
2018
Number of Pages:
819
Format:
Hardcover
Series:
Colloquium Publications 63
Price:
135.00
ISBN:
9781470411046
Category:
Monograph
[Reviewed by
Scott Taylor
, on
06/15/2018
]

Druţu and Kapovich have produced a magisterial introduction to the now well-established, but still relatively recent, field of geometric group theory (GGT). Essentially, GGT studies groups by making a group $G$ act on a metric space $X$ and then using the geometric properties of $X$ to deduce algebraic properties of $G$. Any GGT book must, therefore:

1. Introduce and study geometric properties of metric spaces.
2. Explain how to to convert those properties into properties of a group acting on a metric space, so that we are studying the group itself and not just its action.
3. Relate those geometric properties to algebraic properties.

Geometric Group Theory does all three of these tasks admirably. It will undoubtedly be an essential reference for those working in GGT and related areas in differential geometry and topology. Its wealth of immensely valuable historical commentary is particularly appreciated.

At almost 800 pages of advanced mathematics, the book is suitable for graduate students and professionals. Those without substantial background in geometry, topology, and analysis are advised to begin with more elementary books such as Groups, Graphs, and Trees by John Meier or Office Hours with a Geometric Group Theorist, edited by Clay and Margalit.

Scott Taylor is a geometric topologist at Colby College.