Geometric Group Theory

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9803396Mosher A central theme of geometric group theory is that finitely generatedgroups can be studied geometrically by attempting to classify them up toquasi-isometry. In one part of this project, the investigator, workingjointly with B. Farb of the University of Chicago, will attempt to showthat if G is a finitely generated group that is quasi-isometric to asolvable group, then G is very closely related to some solvable group bysimple algebraic operations, with special attention to fundamental groupsof 3-dimensional solv-manifolds and other polycyclic groups. In anotherpart, joint with M. Sageev, the investigator will work on constructingnew examples of word hyperbolic groups that have high codimension but arenot closely related to arithmetic lattices, a problem posed by M. Gromov.Continuing a long-term effort, the investigator will study the followingweak version of W. Thurston's Hyperbolization Conjecture, namely that ifM is a compact 3-manifold, then the fundamental group of M is either wordhyperbolic or else contains a subgroup isomorphic to a free abeliangroup of rank 2. This part will make use of tools developed in previousjoint work with U. Oertel as well as new tools of ``coarse algebraictopology'' developed recently by geometric group theorists. Continuinganother long-term effort, the investigator will work on showing that if Mis a compact hyperbolic 3-manifold, then only finitely many differentpseudo-Anosov flows are needed to compute Thurston's norm on the secondhomology of M. Groups are the mathematical abstraction of symmetries of geometricobjects. Symmetry patterns of wallpaper, bathroom tiles, and thetilings on the walls of the Alhambra in Spain give examples of Euclideansymmetry groups. Many of Escher's paintings give examples either ofEuclidean or of hyperbolic symmetry groups. This project will coverseveral topics in geometric group theory, the general study of geometriesand their symmetry groups. By matching the large scale properties of ageometry with the large scale properties of its symmetry groups, one canuse geometries to study groups and vice versa, a very fruitful pairingthat has led to many recent mathematical advances. The investigator,working with Benson Farb (U Chicago), will study a special class ofgroups known as ``solvable groups,'' a generalization of Euclideansymmetry groups. Working with his colleague Michah Sageev, he willinvestigate new generalizations of hyperbolic symmetry groups. He willalso continue work on long term efforts to understand 3-dimensionalgeometries and their symmetry groups, as well as to study 3-dimensionaldynamical systems.***