Siegel ’ s Problem in Three Dimensions

We discuss our recent solution to Siegel’s 1943 problem concerning the smallest co-volume lattices of hyperbolic 3-space. Over the last few decades the theory of Kleinian groups—discrete groups of isometries of hyperbolic 3-space—has flourished because of its intimate connectionswith low-dimensional topology and geometry and has been inspired by thediscoveries ofW. P. Thurston. The culminationmust certainly be Perelman’s proof of Thurston’s geometrisation conjecture, which states that compact 3-manifolds can be decomposed canonically into pieces that have Gaven Martin is Distinguished Professor of Mathematics at the NZ Institute for Advanced Study, Massey University, New Zealand. His e-mail address is [email protected]. Work partially supported by the New Zealand Marsden Fund. All article artwork is courtesy of the author. For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1467 geometric structures and that the “generic” piece is hyperbolic. This is an analogue for 3-manifolds of the uniformisation theorem for surfaces and implies, for instance, the Poincaré conjecture. There have been many other recent advances. These include the proofs of the density conjecture of Agol (2004) and of Calegari and Gabai (2006); the ending lamination conjecture of Brock, Canary, and Minsky (2012); the surface subgroup conjecture of Kahn and Markovich (2012); and the virtual Haken conjecture of Agol, Groves, and Manning (2013). While we will not discuss these results here (nor even offer statements of theorems), together they give a remarkably complete picture of the structure of hyperbolic group actions and their quotient spaces—hyperbolic manifolds and orbifolds—in three dimensions. Here we report on joint work with F. W. Gehring and T. H. Marshall [2], [3] solving an old problem of C. L. Siegel. A lattice Γ is the group associated with a tessellation of hyperbolic n-space Hn by a finite hyperbolic volume tile (fundamental domain), and an orbifold is the quotient space Hn/Γ—a hyperbolic manifold Γ should be torsion free. The volumeof the fundamental domain is the volume 1244 Notices of the AMS Volume 63, Number 11 Figure 1. Siegel proved that the (2, 3, 7)-tessellation of the hyperbolic plane is the unique lattice of minimal