Graphs, Trees and Geometric Group Theory

Cornell University, Ithaca NY

Investigators

Abstract

DMS-0204185 Karen Vogtman This project continues to develop a set of ideas which began with an attempt to understand automorphism groups of free groups by representing them as homotopy equivalences of graphs. The space of all such homotopy equivalences, called Outer space, is a contractible space on which the group of outer automorphisms of a free group acts properly discontinuously. Analyzing the quotient of Outer space by this action has yielded a significant amount of algebraic information about this group over the years. More recently, work of Maxim Kontsevich has related the rational homology of this quotient to the homology of a certain infinite-dimensional Lie algebra, consisting of derivations of the free Lie algebra. Kontsevich also defined variations of this infinite-dimensional Lie algebra whose homologies are related to the homology of mapping class groups and to Kontsevich's graph homology, which is a homology theory containing various 3-manifold and knot invariants. The Principal Investigator will use this point of view to obtain new information about invariants of these groups of classical interest to topologists; conversely, topological methods will be applied to obtain new information about these Lie algebras and their homology. In particular, new Lie bi-algebra structures on the chain complexes used to compute graph homology, recently discovered by Jim Conant the the Principal Investivator, will be investivated. Finally, the local geometry of Outer space will be studied. Certain neighborhoods in Outer space are non-positively curved metric spaces, and can be identified with spaces of finite labelled metric trees. Such trees are of interest in many branches of science, including biology, where they appear as evolutionary trees produced from DNA data. Another aim of this project is to determine efficient algorithms for finding geodesics in this space, and to continue from there to develop a program of applying statistical methods to this space of trees. Graphs and trees appear in many contexts in the sciences and in mathematics. In Geometric Group Theory a space of graphs known as Outer space, originally defined by Marc Culler and the Principal Investigator, has been used extensively to study the group of outer automorphisms of a finitely generated free group. In this project the Principal Investigator will continue to study how the geometry of this space is related to algebraic properties of the group of automorphisms. A new perspective on this problem is provided by work of M. Kontsevich; motivated by considerations from physics and symplectic geometry, Kontsevich defined various abstract infinite-dimensional algebraic structures whose invariants are closely related to invariants of Outer space. Finally, small regions of Outer space can be identified with spaces of finite trees. Such trees are used as a tool by evolutionary biologists, and another focus of this project is understanding the local geometry of these regions in a very concrete, algorithmic way which can be applied to problems in molecular phylogenetics.