The research proposed is concerned with the geometric and topological aspects of infinite discrete groups. The proposal divides into two main parts: the large scale geometry of groups and the actions of groups on compact manifolds. Both of these can be viewed as broad generalizations of the celebrated rigidity results of Mostow and Margulis for lattices in Lie groups. A significant amount of research in geometric group theory is devoted to understanding the large scale geometry of lattices in Lie groups. In this vein, recent results of Eskin, Fisher, and the PI establish rigidity for certain solvable Lie groups. The techniques involved are only beginning to be understood, and the PI proposes to develop them further. In particular the PI hopes to apply these techniques to show the invariance of the solvable radical in more general Lie groups and to the quasi-isometric classification of nilpotent Lie groups. The second focal point of the proposal is the study of groups can appear as topological symmetries of compact manifolds. There has been much work on this problem, but the results almost all apply to specific low dimensional manifolds. Even for the homeomorphisms of the circle there are many unresolved questions. The PI proposes studying some classes of higher dimensional manifolds which do have large groups of symmetries on the level of homotopy and to determine whether these symmetries can be realized as groups of homeomorphism.
Roughly speaking, large scale (or coarse) geometry is the study of geometric properties of objects "seen from far away". From this perspective, any bounded object is indistinguishable from a point, and a line of dots is indistinguishable from a solid line. This sort of geometry has been influential recently in many areas of mathematics, notably group theory, topology, and geometric analysis. This research will extend previous work of the PI exploring the large scale geometry of several classes of mathematical objects, both classical geometric spaces and objects only now being viewed in a geometric manner. A particular focus of the proposal is the study of the possible coarse geometries of groups of topological symmetries of relative simple objects built out of familiar spaces, like spheres or other surfaces.
