Geometric Group Theory and the Topology of Aspherical Manifolds

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Abstract Award: DMS-0104026 Principal Investigator: Michael W. Davis This is a proposal for research in geometric group theory focusing on Coxeter groups, Artin groups, and mapping class groups of surfaces. The main problems to be addressed are the following. (1) For which Coxeter groups is the Coxeter diagram uniquely determined by the group? For which Coxeter groups is the fundamental generating set uniquely determined (up to conjugation) by the group? For which Coxeter groups is the outer automorphism group finite? (2) Are all Artin groups linear groups? (3) Find a formula for the cohomology with compact supports of a building. Likewise for the Salvetti complex of an Artin group. (4) Determine the l^2 Betti numbers of cubical manifolds associated to right-angled Coxeter groups. Do they vanish outside the middle dimension? This would imply the Flag Complex Conjecture concerning triangulations of odd-dimensional spheres and has implications for graph embeddings. (5) Develop a theory of mock reflection groups. This is a class of groups similar to Coxeter groups which arise as transformations of blow-ups of hyperplane arrangements. (6) Is the Torelli subgroup of the mapping class groups of a surface of genus at least three finitely generated? (7) Can a word hyperbolic group be the fundamental group of a surface by surface bundle? Must all finitely presented non word hyperbolic groups contain a Baumslag-Solitar group or an abelian group of rank two? Group theory arises from the study of symmetries of an object. When this object has an interesting geometric structure, one can use geometric techniques to better understand the group of symmetries. This project involves the study of certain families of groups which arise in a broad range of mathematical and physical contexts, such as the study of crystal structures and the intertwining of DNA. These groups are associated to rich and beautiful geometric structures which lend themselves to the techniques of geometric group theory.