2013 Veblen Prize Ian Agol and Daniel Wise received the 2013 quasiconvex subgroup of a hyperbolic group is Oswald Veblen Prize in Geometry at the 119th An- separable. nual Meeting of the AMS in San Diego, California, In April 2012 Agol posted an article to the arXiv in January 2013. announcing a proof of a conjecture of Daniel Wise implying both Waldhausen’s virtual Haken conjec- Citation for Ian Agol ture and Thurston’s virtual fibering conjecture. The 2013 Oswald Veblen Prize in Geometry is The proof involves a generalization of the methods awarded to Ian Agol for his many fundamen- of these papers, clever new ideas, and it makes tal contributions to hyperbolic crucial use of results of Wise and joint work of geometry, 3-manifold topology, Haglund and Wise. The resolution of these conjec- and geometric group theory. In tures is a milestone achievement in the theory of particular, Agol is cited for the 3-dimensional manifolds. following papers: (with P. Storm and W. P. Thur- Biographical Sketch ston) “Lower bounds on volumes Ian Agol was born in Hollywood, California, in 1970 of hyperbolic Haken 3-manifolds” and received his Ph.D. at the University of Califor- with an appendix by Nathan Dun- nia San Diego in 1998 under the supervision of field, Journal of the American Michael Freedman. He held postdoctoral positions Mathematical Society 20 (2007), at UC Davis and the University of Melbourne before no. 4, 1053–1077. teaching at the University of Illinois at Chicago. “Criteria for virtual fibering”, He moved to UC Berkeley in 2007. He received a Journal of Topology 1 (2008), Guggenheim fellowship in 2005 and spoke at the Ian Agol no. 2, 269–284. International Congress of Mathematicians (ICM) in (with D. Groves and J. F. Man- Madrid (2006). ning) “Residual finiteness, QCERF and fillings of He shared the 2009 Clay Research Award with hyperbolic groups”, Geometry and Topology 13 Danny Calegari and David Gabai for the solution (2009), no. 2, 1043–1073. of the Marden tameness conjecture. He received In 2004 Agol (and independently Calegari and the 2012 Senior Berwick Prize for his cited paper, Gabai) proved the Marden tameness conjecture, “Criteria for virtual fibering”. which led to many important results in hyper- bolic geometry, among them the Ahlfors measure Response conjecture. It is an honor to share the 2013 Oswald Veblen Agol subsequently proved a number of impor- Prize in Geometry with Dani Wise. tant results on volumes of hyperbolic 3-manifolds, I owe a debt of gratitude to my family for their including an important inequality, found with support, especially my mother, who made sure Nathan Dunfield in the first cited paper, relating that I had a top-notch education and encouraged the volume of a closed hyperbolic 3-manifold with me to pursue a career in a field I am passionate that of a drilled manifold and the tube radius of about, and my wife, Michelle, for her support. I also the drilled geodesic. This played an important role share this award in spirit with my collaborators, in the proof (by Gabai, Meyerhoff, and Milley) that especially Nathan Dunfield, Peter Storm, Daniel the Weeks manifold is the unique lowest volume Groves, Jason Manning, and the late Bill Thurston, closed orientable hyperbolic 3-manifold. with whom I collaborated on the papers mentioned In the second paper Agol found a key group in the citation. My research on the virtual Haken theoretic criterion for a closed irreducible 3-mani- conjecture was influenced heavily by the work of fold to have a finite sheeted covering space that Dani Wise and his collaborators Nicolas Bergeron, fibers over the circle. With Daniel Groves and Jason Tim Hsu, Michah Sageev, and especially Frédéric Manning, he showed in the third paper that if all Haglund. Without Wise’s insights, this approach to hyperbolic groups are residually finite, then any the conjecture would have been impossible, and his vision completely changed my perspective on DOI: http://dx.doi.org/10.1090/noti973 the problem. Another key contribution was made 494 NOTICES OF THE AMS VOLUME 60, NUMBER 4 by Jeremy Kahn and Vlad Markovic on the surface (with F. Haglund) “Spe- subgroup problem. The subject of hyperbolic cial cube complexes”, groups and CAT(0) cube complexes, introduced by Geometric and Func- Misha Gromov, has reached a mature state, which tional Analysis 17 (2008), I think is now made evident by the first Veblen no. 5, 1551–1620. Prize given to a geometric group theorist after (with F. Haglund) “A com- Gromov himself. bination theorem for special In the 1980s, Bill Thurston laid out his vision cube complexes”, Annals of for three-dimensional topology and Kleinian Mathematics 176 (2012), no. groups. Over the past thirty years, his vision has 3, 1427–1482. In the first of been validated, with such milestones achieved as these papers, Wise and Ha- the geometrization theorem, the ending lamina- glund introduced the concept tion theorem, the tameness theorem, the density of a “special cube complex”. theorem, and now the virtual fibering theorem. The The fundamental groups of Daniel Wise virtual Haken theorem (originating as a question such complexes have various of Waldhausen) is a purely topological statement; remarkable properties. For example, they embed however, its resolution is in some sense mostly into right-angled Artin groups and their quasi- geometrical and depends on many developments convex subgroups are separable. Generalizing the from outside the field, including PDEs, Riemannian methods of the two papers of Wise cited above, and Alexandrov geometry, dynamics, representa- Wise and Haglund show that a group obtained tion theory, and geometric group theory. from two virtually special hyperbolic groups that I hope that 3-manifolds and Kleinian groups are amalgamated along quasiconvex malnormal will give back equally to these and other subjects subgroups is virtually special. In subsequent work in the future. Moreover, I think the most impor- Wise used this to prove that the fundamental tant project for the future is to make connections groups of Haken hyperbolic 3-manifolds are virtu- between this highly developed area of geometric ally special. 3-manifold topology and other approaches, such Beyond what is cited in these papers, Wise made as various flavors of Floer homology, topological the far-reaching conjecture that a Gromov hyper- quantum field theory, knot theory and 4-mani- bolic group that acts cocompactly and properly on folds, and other types of geometric structures on a CAT(0) cube complex is the fundamental group of 3-manifolds. virtually special cube complexes. Ian Agol proved Citation for Daniel Wise this conjecture making essential use of the technol- The 2013 Veblen Prize in Geometry is awarded to ogy of Wise and his collaborators. Combined with Daniel Wise for his deep work establishing sub- the work of Jeremy Kahn and Vladimir Markovic, group separability (LERF) for a wide class of groups and with the work of Michah Sageev, this gives a and for introducing and developing with Frédéric positive solution to the long-standing Waldhausen Haglund the theory of special cube complexes, virtual Haken conjecture. which are of fundamental importance for the to- Biographical Sketch pology of 3-dimensional manifolds. In particular Wise is cited for the following papers: Daniel Wise was born in 1971 and grew up in the “Subgroup separability of graphs of free groups New York City area. He received a B.A. in 1991 from with cyclic edge groups”, Quarterly Journal of Yeshiva College and a Ph.D. in 1996 from Princeton Mathematics 51 (2000), no. 1, 107–129. A sub- University. After stints at UC Berkeley, Cornell Uni- group is separable if it is the intersection of all versity, and Brandeis University, he joined McGill the finite index subgroups that contain it. This University, where he has been teaching since 2001. is an important condition in 3-manifold topology Dani lives in Montreal with his wife, Yael, and their since up to homotopy an immersed surface in a four children. Their house is full of music, art, closed irreducible 3-manifold lifts to an embedded laughter, and sleep deprivation. surface in some finite sheeted cover if the immer- sion induces an isomorphism of the fundamental Response group of the surface with a separable subgroup of I am grateful to be the recipient, together with the fundamental group of the 3-manifold. Ian Agol, of the Oswald Veblen Prize of the AMS. “Residual finiteness of negatively curved poly- I share my part of this recognition with my coau- gons of finite groups”, Inventiones Mathematicae thors Nicolas Bergeron, Chris Hruska, Tim Hsu, 149 (2002), no. 3, 579–617. This paper is represen- Michah Sageev, and especially Frédéric Haglund. tative of a long arc of work in which Wise describes This is a welcome opportunity to thank Jonathan powerful and ingenious criteria for showing that a Sondow and Sylvain Cappell, who launched me into group is residually finite or, more generally, that mathematics, Martin Bridson, who initiated my all of its quasiconvex subgroups are separable. interest in geometric group theory, my colleagues APRIL 2013 NOTICES OF THE AMS 495 and students at McGill, my friends and family, and committee. For the 2010 prize, the members of my wife, Yael. the selection committee were Tobias H. Colding, I was attracted to residual finiteness at the very William G. Dwyer, and Michael J. Hopkins. beginning of my career—and latched on to the Previous recipients of the Veblen Prize are: idea of a “clean complex” in 1993. From that point Christos D. Papakyriakopoulos (1964), Raoul H. onward I progressively nurtured a belief that a vari- Bott (1964), Stephen Smale (1966), Morton Brown ant of this was a key toward understanding basic and Barry Mazur (1966), Robion C.