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3-manifold groups. (English) Zbl 1326.57001
EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-154-5/pbk). xiv, 215 p. (2015).
In the first half of the last century, combinatorial group theory developed motivated by and in close connection with low-dimensional topology; apart from free groups (and knot groups), the main prototype class of groups considered were the fundamental groups of closed surfaces (surface groups) which led Dehn to formulate the isomorphism, word and conjugacy problems for finite presentations of groups. The first classical period ended around 1980, after the results of Haken on surfaces in 3-manifolds, Papakyriakopoulos’ work on the Dehn Lemma and the loop and sphere theorems, Waldhausen’s results on Haken 3-manifolds (leading in particular to solutions of the word and isomorphism problems for their fundamental groups), and the work of Jaco-Shalen and Johannson on torus decompositions of 3-manifolds (JSJ-decompositions). All these results have strong impacts on the fundamental groups of closed 3-manifolds (3-manifold groups); nevertheless the class of 3-manifold groups and their possible subgroups remained mysterious; in particular, not much was known for non-Haken 3-manifolds with infinite fundamental group (e.g., the existence of surface subgroups), and on 3-manifolds with finite fundamental group (including the Poincaré conjecture).
As is well-known, the somewhat stagnating and isolated situation (due also to the lack of strong enough methods, e.g. for the Poincaré-conjecture) was revolutionized by Thurston by introducing geometric methods and structures in a systematic way, formulating in 1982 his famous and most influential list of 24 problems on 3-manifolds and Kleinian groups [W. P. Thurston, Bull. Am. Math. Soc., New Ser. 6, 357–379 (1982; Zbl 0496.57005); see also the recent comments and updates on Thurston’s problem list by J. P. Otal, Jahresber. Dtsch. Math.-Ver. 116, No. 1, 3–20 (2014; Zbl 1301.00035)]; the development culminated, after the geometrization of Haken 3-manifolds due to Thurston, in the proof of the geometrization conjecture for general closed 3-manifolds by Perelman (including the Poincaré-conjecture, and relying heavily on analytic methods from differential geometry). It became clear then that the main class of 3-manifolds are the hyperbolic 3-manifolds, in particular their fundamental groups are Kleinian groups acting by isometries on hyperbolic 3-space, and hence share many properties with linear groups; as a first group-theoretical consequence, 3-manifold groups are residually finite.
Problems 5-14 of Thurston’s list are on Kleinian groups (among these the tameness conjecture, the ending lamination conjecture and the density conjecture); due to deep work of Agol, Calegari-Gabai, Minsky and Brock-Canary-Minsky these problems are all resolved now (in a second step following the geometrization).
Problems 15-18 of the list are on 3-manifold groups and their subgroups: subgroup separability of 3-manifold groups (or LERF-property, standing for locally extended residually finite, a generalization of residually finite), the (related) virtual Haken conjecture and the question whether every hyperbolic 3-manifold has a finite cover fibering over the circle, or one with positive first Betti number. In a third and quite recent step, also these problems are all resolved now, due to work of Wise, Agol and Kahn-Markovic, and these results and their group-theoretical implications are the central topic of the present monograph (written in large parts as a survey on these developments and on the implications on 3-manifold groups).
The nine chapters of the book are as follows: 1. Decomposition theorems; 2. The classification of 3-manifolds by their fundamental groups; 3. 3-manifold groups after geometrization; 4. The work of Agol, Kahn-Markovic, and Wise; 5. Consequences of the virtually compact special theorem; 6. Subgroups of 3-manifold groups; 7. Open questions; finally, a Bibliography of 60 pages and an Index of 5 pages.
In a far-reaching generalization to 3-manifolds of an approach used by Peter Scott for surfaces, Daniel Wise developed the notion of cube complex, on various levels of specialization, and of cubulated group (a cube complex is a cell complex whose cells are Euclidean cubes, with isometries as attaching maps; related is also the concept of right-angled Artin group). Then any cocompact Kleinian group turns out to be cubulated; decisive here (in the case of non-Haken 3-manifolds) is the recent solution of Kahn and Markovic on the surface subgroup conjecture, or on the existence of many immersed surfaces in hyperbolic 3-manifolds, allowing the construction of such structures. The last step to the solution of problems 15-18 was then done by Agol (see also the recent survey-article by S. Friedl [Jahresber. Dtsch. Math.-Ver. 116, No. 4, 223–241 (2014; Zbl 1305.57003)]). All of this has a strong taste of geometric group theory which, after Gromov, enlarged the classical combinatorial group theory; important concepts are right-angled Artin groups, convex, non-positively curved subgroups and, for Kleinian groups, the dichotomy between geometrically finite (quasi-Fuchsian) and geometrically infinite (virtual fiber) subgroups (finitely generated, of infinite index).
“The goal of this book is to fill what we perceive as a gap in the literature, and to give an extensive overview of results on fundamental groups of compact 3-manifolds with a particular emphasis on the impact of the Geometrization Conjecture of Thurston and its proof by Perelman, the Tameness Theorem of Agol and Calegari-Gabai, and the Virtually Compact Special Theorem of Agol, Kahn-Markovic and Wise. Our approach is to summarize many of the results in several flowcharts and to provide detailed references for each implication appearing in them. We will mostly consider fundamental groups of 3-manifolds which are either closed or have toroidal boundary.”
To conclude, I think that this is a very useful guide to some of the most exiting recent developments in low-dimensional topology and geometric group theory, even more to appreciate since many of the results are quite recent, various not even yet published, and hence still in considerable fluctuation. It is the first text presenting these results and their group-theoretical consequences in a systematic way (the result is a kind of handbook on 3-manifolds and their fundamental groups); concerning proofs, clearly much work still has to be done in order to assimilate and present the often quite long and technical proofs on a more accessible conceptual level – but this is another story which still has to be written.

##### MSC:
 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57M50 General geometric structures on low-dimensional manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010) 20F65 Geometric group theory
##### MathOverflow Questions:
Quantitative word problem for 3-manifold groups
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