Asymptotic Properties of 3-Manifolds and Their Fundamental Groups

Brigham Young University, Provo UT

Investigators

Abstract

Abstract Award: DMS-0104030 Principal Investigator: James W. Cannon We attempt to resolve the hyperbolic case of Thurston's Geometrization Conjecture for 3-dimensional manifolds. Among the many possible approaches, we choose to study the asymptotic recursive properties of the fundamental group of the manifold. We concentrate on the asymptotic shingling patterns at infinity defined by the group and seek methods for proving that such patterns do or do not satisfy the necessary and sufficient conformality axiom which we introduced in our earlier studies. We study this problem in three contexts: (1) We study closed 3-manifolds with Gromov-hyperbolic group, with emphasis on the examination of concrete examples constructed by our new method of twisted-face-pairings; (2) We study general subdivision or local-replacement rules in the plane, where we have more freedom in constructing examples with special properties; and (3) We study branched coverings of the 2-sphere by the 2-sphere, where the corresponding problem is to some extent already solved by means of Thurston's combinatorial characterization of rational maps. It is in the third context that the connection with classical Teich- mueller theory becomes apparent; we are seeking an appropriate version of Teichmueller theory for our setting. In all of these contexts, we use the circle-packing programs of Ken Stephenson, the automatic group programs of Epstein, Holt, and Rees, and the program SnapPea of Jeff Weeks in conjunction with programs of the proposer and his coworkers to construct, geometrically optimize, and explore the patterns being studied. William P. Thurston has supplied us with a powerful conjectural picture of the spaces of 3-dimensional mathematics, the 3-dimensional manifolds. Thurston's Geometrization Conjecture is the most important unresolved problem in low dimensional topology, even if one sets aside the case of spherical geometry where the conjecture implies the famous "million dollar" Poincare Conjecture. Thurston suggests that every 3-manifold can be divided in an intrinsic manner into pieces, each of which is modelled on one of eight natural geometries. Within each piece, one can apply well-understood algebraic and geometric techniques to derive properties of the manifold. This project seeks to resolve the generic case of the Thurston Conjecture, namely the case of hyperbolic geometry. The technique employed is to maximally unwind the manifold, that is, take its universal cover, and study the asymptotic properties of this cover. The cover can be studied combinatorially almost as a growing cellular organism as a plant or animal might be studied by a cell biologist. The cover can be studied computationally as a cellular automaton might be studied by a computer scientist. The cover can be studied analytically by methods of discrete dynamical systems or differential equations or conformal mapping.