Research in Geometry and Topology

University Of Utah, Salt Lake City UT

Investigators

Abstract

The proposal centers around 3 topics. The first is about questions in group theory motivated by the first order predicate calculus in logic. The questions involve e.g. understanding the sets of homomorphisms of a fixed group into a free group that extend to another fixed group. The second topic concerns the topology of moduli space of curves and of the Torelli subgroup of a mapping class group. The idea is to see if homology can be computed using a suitably chosen Morse function on the moduli space, and on the quotient of Teichmuller space by the Torelli group, respectively. The Morse function is geometrically defined -- it is the systole, i.e. the length of the shortest curve (or the homological version of this in the case of Torelli groups). The last topic is on the quasi-isometric rigidity of right-angled Artin groups. The subject has seen some great advances in the last 15 years or so, but this particular class of groups raises many issues not addressed before. The subject of the proposal is the study of topological and geometric properties of several objects that naturally appear in mathematics. For example, consider the collection of all possible "shapes" of a surface with a fixed number of holes. This collection forms a space, called moduli space of curves. The calculation of the "number of holes" for this space has been recently completed by a topological tour de force. A different method for this calculation is proposed here, one that would also give additional information. To describe the method, consider the following analogy. When water is slowly poured into a normal glass, the surface of water in the glass will always be a disk. By contrast, if water is poured into a glass with a hollow handle, the surface of water will consist of two disks once water reaches the handle. Thus, by examining the shape of the surface of water we can tell if a glass has a handle or not. Topological spaces, such as moduli space of curves, can be analyzed similarly. A particular way of "pouring water" is proposed here.