Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston's vision (Q2864922)

This is a survey on the proof by Agol of the virtual fibration conjecture, namely that every closed hyperbolic three-manifold has a finite sheeted covering that fibers over the circle. Agol's theorem answers several of the question raised by \textit{W. P. Thurston} in his celebrated survey [Bull. Am. Math. Soc., New Ser. 6, 357--379 (1982; Zbl 0496.57005)]. In particular it establishes a conjecture of Waldhausen: every irreducible compact three-manifold is virtually Haken.NEWLINENEWLINEThe paper illustrates the strong connections between geometric group theory and hyperbolic three-manifolds, as the proof relies on cube complexes of non-positive curvature. The proof builds on a theorem of \textit{J. Kahn} and \textit{V. Markovic} on the existence of almost geodesic immersed surfaces in any hyperbolic three-manifold [Ann. Math. (2) 175, No. 3, 1127--1190 (2012; Zbl 1254.57014)]. By means of Sageev construction applied to these surfaces, for any closed hyperbolic three-manifold Bergeron and Wise constructed a compact cube complex of non-positive curvature with the same fundamental group [\textit{D. G. Green}, Pac. J. Math. 57, 141--152 (1975; Zbl 0279.20051)]. In addition, \textit{F. Haglund} and \textit{D. T. Wise} had developed the theory of special cube complexes [Ann. Math. (2) 176, No. 3, 1427--1482 (2012; Zbl 1277.20046)], with the aim to promote immersions to embeddings, and that combined with a paper of \textit{I. Agol} [J. Topol. 1, No. 2, 269--284 (2008; Zbl 1148.57023)] gives a sufficient condition for virtual fibering over the circle. The key step is the proof by Agol of a conjecture of Wise: every compact cube complex of non-positive curvature with hyperbolic fundamental group is virtually special [\textit{I. Agol}, Doc. Math., J. DMV 18, 1045--1087 (2013; Zbl 1286.57019)]. As a consequence, every closed hyperbolic three-manifold has a finite sheeted covering that fibers over the circle. In addition, its fundamental group is large and LERF. Being large for a group means that it has a finite index subgroup that surjects onto a free group of rank two, and LERF stands for locally extended residually finite (every finitely generated subgroup can be separated).NEWLINENEWLINEAfter a quick historical motivation, the paper under review introduces cube complexes of non-positive curvature and explains how they are used in the proof. In particular, it explains Sageev's construction and the key notion of special cube complexes. Other ingredients that play a role in the proof are also explained here: a Dehn filling theorem for groups, several results of Wise on special complexes, and the Agol-Groves-Manning weak separation theorem (which is an appendix to Agol's paper). Finally, after recalling the work of Kahn-Markovich and Bergeron-Wise, it explains the proof of Wise's conjecture.NEWLINENEWLINEThe survey is well written and very clear. As the title suggests, it shows the strong and deep ties between geometric group theory and three-manifold topology, and it is a nice tribute to Thurston.