Research in Geometry and Topology

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9971404Bestvina The goal of this project is to understand 3-dimensional Poincare duality groups and pairs, subgroups of mapping class groupsand automorphisms of free groups, complex-hyperbolic surfaces,polygons in symmetric spaces, line arrangements, Artin groups, anddimension functions on groups. The project focuses in part on the structure of symmetries ofobjects naturally arising in geometry and algebra. The ancient Greeksunderstood symmetries of Platonic solids, e.g., a cube or adodecahedron. Platonic solids are positively curved, and they have afinite number of symmetries. At the opposite end of the spectrum arenegatively curved objects, such as the Lobachevsky hyperbolic plane(featured frequently in Escher paintings), and these typically haveinfinitely many symmetries. Consequently, the structure of thesesymmetries is both more interesting and harder to understand. Thesymmetries of an object form a ``group.'' One way to measure the``size'' of a group, even when the group is infinite, is to talk aboutits ``dimension.'' For example, the dimension of the group ofsymmetries of the hyperbolic plane, depicted in the Escher drawingmentioned above, is two, reflecting the nature of the plane. Ingeneral, the concept of dimension depends on the background choice ofnumbering system (coefficient ring). The project will study thisdependence. Yet another part of the project is the study of polygons.These seemingly simple objects in the Euclidean plane and space becomeincreasingly more complex as one considers more complicated spaces. Oneof the new phenomena is that an edge of a polygon may be determined notjust by its length but also by its ``direction.'' In the case ofordinary space, if one fixes the origin of an edge, one can rotate thesecond end to attain an arbitrary direction. A Euclidean triangle isdetermined by its three side-lengths. To construct such a triangle itis enough to satisfy three ``triangle inequalities.'' These propertiesno longer hold in more general spaces, which makes study of polygonsthere much more interesting and complicated.***