Asymptotic Properties of 3-Manifolds and Their Fundamental Groups

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AbstractAward: DMS-0104030Principal Investigator: James W. CannonWe attempt to resolve the hyperbolic case of Thurston'sGeometrization Conjecture for 3-dimensional manifolds. Among themany possible approaches, we choose to study the asymptoticrecursive properties of the fundamental group of the manifold. Weconcentrate on the asymptotic shingling patterns at infinitydefined by the group and seek methods for proving that suchpatterns do or do not satisfy the necessary and sufficientconformality axiom which we introduced in our earlier studies. Westudy this problem in three contexts: (1) We study closed3-manifolds with Gromov-hyperbolic group, with emphasis on theexamination of concrete examples constructed by our new method oftwisted-face-pairings; (2) We study general subdivision orlocal-replacement rules in the plane, where we have more freedomin constructing examples with special properties; and (3) Westudy branched coverings of the 2-sphere by the 2-sphere, wherethe corresponding problem is to some extent already solved bymeans of Thurston's combinatorial characterization of rationalmaps. It is in the third context that the connection withclassical Teich- mueller theory becomes apparent; we are seekingan appropriate version of Teichmueller theory for our setting. Inall of these contexts, we use the circle-packing programs of KenStephenson, the automatic group programs of Epstein, Holt, andRees, and the program SnapPea of Jeff Weeks in conjunction withprograms 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 conjecturalpicture of the spaces of 3-dimensional mathematics, the3-dimensional manifolds. Thurston's Geometrization Conjecture isthe most important unresolved problem in low dimensionaltopology, even if one sets aside the case of spherical geometrywhere the conjecture implies the famous "million dollar" PoincareConjecture. Thurston suggests that every 3-manifold can bedivided in an intrinsic manner into pieces, each of which ismodelled on one of eight natural geometries. Within each piece,one can apply well-understood algebraic and geometric techniquesto derive properties of the manifold. This project seeks toresolve the generic case of the Thurston Conjecture, namely thecase of hyperbolic geometry. The technique employed is tomaximally unwind the manifold, that is, take its universal cover,and study the asymptotic properties of this cover. The cover canbe studied combinatorially almost as a growing cellular organismas a plant or animal might be studied by a cell biologist. Thecover can be studied computationally as a cellular automatonmight be studied by a computer scientist. The cover can bestudied analytically by methods of discrete dynamical systems ordifferential equations or conformal mapping.