A central theme of geometric group theory is that finitely generated groups can be studied geometrically by attempting to classify them up to quasi-isometry. In one part of this project, the investigator, working jointly with B. Farb of the University of Chicago, will attempt to show that if G is a finitely generated group that is quasi-isometric to a solvable group, then G is very closely related to some solvable group by simple algebraic operations, with special attention to fundamental groups of 3-dimensional solv-manifolds and other polycyclic groups. In another part, joint with M. Sageev, the investigator will work on constructing new examples of word hyperbolic groups that have high codimension but are not closely related to arithmetic lattices, a problem posed by M. Gromov. Continuing a long-term effort, the investigator will study the following weak version of W. Thurston's Hyperbolization Conjecture, namely that if M is a compact 3-manifold, then the fundamental group of M is either word hyperbolic or else contains a subgroup isomorphic to a free abelian group of rank 2. This part will make use of tools developed in previous joint work with U. Oertel as well as new tools of ``coarse algebraic topology'' developed recently by geometric group theorists. Continuing another long-term effort, the investigator will work on showing that if M is a compact hyperbolic 3-manifold, then only finitely many different pseudo-Anosov flows are needed to compute Thurston's norm on the second homology of M. Groups are the mathematical abstraction of symmetries of geometric objects. Symmetry patterns of wallpaper, bathroom tiles, and the tilings on the walls of the Alhambra in Spain give examples of Euclidean symmetry groups. Many of Escher's paintings give examples either of Euclidean or of hyperbolic symmetry groups. This project will cover several topics in geometric group theory, the general study of geometries and their symmetry groups. By matching the large scale properties of a geometry with the large scale properties of its symmetry groups, one can use geometries to study groups and vice versa, a very fruitful pairing that has led to many recent mathematical advances. The investigator, working with Benson Farb (U Chicago), will study a special class of groups known as ``solvable groups,'' a generalization of Euclidean symmetry groups. Working with his colleague Michah Sageev, he will investigate new generalizations of hyperbolic symmetry groups. He will also continue work on long term efforts to understand 3-dimensional geometries and their symmetry groups, as well as to study 3-dimensional dynamical systems. ***
