3-Manifold Groups

Associated to every “nice” topological space is the fundamental group of the space. It is the group whose elements are paths beginning and ending at a fixed basepoint, with two paths being equivalent if one can be continuously deformed into the other without moving the endpoints. The group operation is concatenation: travel around one path, then travel around the other. Homeomorphic spaces have isomorphic fundamental groups. A fundamental group is typically infinite and non-abelian. Compact manifolds (possibly with boundary) are very nice topological spaces which play a central role in mathematics and their fundamental groups are key to understanding them.

The story of manifold fundamental groups is a bit Goldilocksy: sometimes the subject is too difficult; sometimes it’s too easy; and rarely is it “just right.” In high dimensions the fundamental group is too hard to understand: for every \(n \geq 4\), every finitely presented group is the fundamental group of some \(n\)-dimensional manifold. As group theory is provably (!) a difficult subject, high-dimensional manifolds are typically studied by imposing constraints on the fundamental group. On the other hand, the fundamental groups for 1-dimensional manifolds are easy to understand. The circle and the line segment are the only 1-dimensional manifolds and (up to isomorphism) their fundamental groups are the trivial group and the integers. This leaves the fundamental groups of 2- and 3-dimensional manifolds.

Two-dimensional manifolds are called surfaces, and their fundamental groups are called surface groups. A typical task in beginning algebraic topology is to write down a presentation for surface groups. This is akin to the theorem which classifies compact surfaces up to homeomorphism solely in terms of the orientability, the genus, and the number of boundary components. Although surface groups have been classified for a long time, there is still interesting work being done on them (especially on their outer automorphism groups.)

Understanding the fundamental groups of 3-dimensional manifolds poses a different sort of challenge. For example, the infamous Poincaré Conjecture is the statement that of all compact 3-manifolds without boundary, only the 3-dimensional sphere has trivial fundamental group. In the 1980s, Bill Thurston reunited 3-dimensional topology and geometry by formulating (and proving an important special case of) his Geometrization Conjecture. In 2003, Grigori Perleman proved both the Poincaré Conjecture and Geometrization Conjecture. The resolution of both conjectures had far-reaching implications for 3-manifolds, including for the understanding of their fundamental groups.

The Geometrization Conjecture (now theorem) and the work of Thurston and others, showed that of all 3-dimensional manifolds, those that admit a complete hyperbolic Riemannian metric are the most interesting. In the 1982 Bulletin of the AMS article where he poses the Geometrization Conjecture, Thurston also asks 24 questions concerning hyperbolic 3-manifolds and their fundamental groups. Those, apart from the Geometrization Conjecture, which have attracted the most attention are questions 16, 17, and 18. Each of these concerns the “virtual” properties of the fundamental groups of hyperbolic 3-manifolds; that is the properties belonging to some finite index subgroup. All three questions were, at some point, promoted from questions to conjectures: the Virtually Haken Conjecture, the Virtual Positive Betti Number Conjecture, and the Virtual Fibering Conjecture. The last of these asks whether or not every compact hyperbolic 3-manifold without boundary and with infinite fundamental group has a finite cover which is fibered; i.e. which is constructed by taking a compact surface without boundary times an interval and then gluing the boundary components together by some homeomorphism. In 2012, Ian Agol (building on work of others) proved the Virtually Fibering Conjecture and, as a consequence, the Virtually Haken and Virtual Positive Betti Number Conjectures.

The resolution of Thurston’s Conjectures has a large number of consequences for the understanding of the fundamental groups of 3-dimensional manifolds. The purpose of the book under review is to give a survey of the state-of-the-art knowledge concerning 3-manifold groups. The book assumes a fair amount of background in both (infinite) group theory and 3-manifold theory, but reminds the reader of essential definitions. It is very clearly written and includes citations to well-over 1000 other sources. The book pays particular attention to recent developments and contains a brief overview of the machinery from geometric group theory and hyperbolic geometry used by Agol and others in their work on the Virtual Fibering Conjecture. The charts concerning the relationship between various geometric and group theoretic ideas are especially useful. The last chapter of the book poses a number of open problems, showing that the reports of the death of 3-manifold theory are greatly exaggerated. 3-Manifold Groups will be the go-to guide on fundamental groups of 3-manifolds for a long time to come.

Scott Taylor is a knot theorist and 3-manifold topologist who rarely hangs out with groups. He is an associate professor at Colby College.