Topics in the Geometry of Groups and Complexes

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AbstractAward: DMS-0906962Principal Investigator: Noel BradyThe long-term objective of this research agenda is to understandthe large scale geometry of groups and complexes. Specificobjectives include the analysis of higher dimensional fillinginvariants of groups, higher filling invariants at infinity(higher divergence), and investigating the structure ofasymptotic cones of finitely presented groups. Other specificobjectives include the use of combinatorial Morse theory inunderstanding virtual fibering in certain classes of groups, andin developing an approach to the relator gap problem. Theprincipal investigator also intends to compare non-positivecurvature methods and topological methods (using stablecommutator length) for proving the existence of surface subgroupsin various classes of hyperbolic groups.Groups are used by mathematicians to study symmetry. A group isjust a collection of symmetries of an object. Examples includethe geometric symmetries of a wallpaper pattern, or of a crystalstructure, or of an Escher painting, or the algebraic symmetriesassociated to roots of polynomials. Mathematicians have studiedgroups intensively as abstract algebraic objects since the 19thcentury. In the 1980's M. Gromov proposed that we consider groupsas geometric objects, and began to derive deep connectionsbetween the geometric and the algebraic properties of groups. Onetheme which emerged from Gromov's work is that the geometricproperties which have deep algebraic consequences are not localproperties, but rather coarse or "large scale". For example, andinfinite ladder and an infinite straight line are not locallyalike, but are large scale alike, and their symmetry groups willhave many algebraic similarities. The PI investigates large scaleversions of isoperimetric problems (area versus perimeter lengthproblems) in groups, and also the geometry of coarsely negativelycurved groups. These investigations help deepen our understandingof the nature of symmetry.