Mathematical Sciences: Topics in Low-Dimensional Topology and Geometric Group Theory

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9504946 Mosher This research project is focussed on negative curvature in the sense of Gromov. The principal investigator (working jointly with U. Oertel) has proved a "Lamination Theorem" which says that for a compact cell-complex X, exactly one of two alternatives holds: either X is negatively curved in the sense of Gromov, or there is a 2-dimensional lamination mapping to X with certain properties, namely the lamination has a transverse measure of Euler characteristic zero, and the map from the universal cover of a leaf to X is least area. The primary aim of this project is to use the Lamination Theorem to investigate the "weak hyperbolization conjecture" for a 3-manifold M, which says that M is negatively curved in the sense of Gromov if and only if the fundamental group of M has no Z+Z subgroup. The strong form of this conjecture has been one of the greatest challenges in 3-manifold theory. The weak conjecture should be a significant step in proving the strong conjecture, in light of recent work of Cannon on characterizing hyperbolic 3-manifold groups, and work of Gabai on the topological rigidity problem for hyperbolic manifolds; if these various projects can be completed, then the hyperbolization conjecture would be proved. This research project is also concerned with several other topics, in particular: understanding properties of automatic and biautomatic groups; the geometric group theory of mapping class groups; constructions of and properties of pseudo-Anosov flows on 3-manifolds. Group theory, the mathematical study of symmetry invented by E. Galois in the 1700's, has for much of its history been a subject of abstract algebra, despite its geometric origins. Starting with work of M. Gromov, J. Cannon, W. Thurston and others in the 1970's and 1980's, the newly emergent field of geometric group theory has returned group theory to its origins. The basic problem, proposed by Gromov in his 1983 address to the International Congress of Math ematics, is to take a "group" (an abstractly described collection of symmetries) and to understand its geometry. As a particular case of Gromov's program, suppose that S is a 3-dimensional space, much like the space that we inhabit, and suppose that G is a certain group of symmetries of S; in this situation Thurston conjectured in the late 1970's that the geometry of G should fall into a list of specific classes. In most cases Thurston's conjecture says that G should have "hyperbolic" geometry, a type of geometry that describes how 3-dimensional space fits into 4-dimensional space-time in relativity theory. The main thrust of the current research project is an attempt to prove Thurston's "hyperbolization conjecture." Recent work of D. Gabai, J. Cannon, and the investigator wih U. Oertel suggests a three-step approach to proving the hyperbolization conjecture, and this project is concerned with one step, the so-called "weak hyperbolization conjecture." While this is an ambitious project, the investigator expects it to be productive; if it is completely successful, and if the other two steps are finished as the work of Cannon and of Gabai suggests, then the hyperbolization conjecture will be proved. ***