Geometric Group Theory

Abstract Award: DMS-0103208 Principal Investigator: Lee Mosher The motivating theme of geometric group theory is that finitely generated groups can be studied using topological and geometric methods. Starting around 1980, Gromov proposed classifying groups geometrically using the relation of quasi-isometry. He demonstrated that the quasi-isometry class of a group G can often be described explicitly in simple algebraic or geometric terms, a process now referred to as ``quasi-isometric rigidity'' or ``QI-rigidity'' for the group G. The first part of this project will be to investigate QI-rigidity problems for graphs of surface groups and of abelian groups. An initial focus will be those graphs of groups with the simplest algebraic structure, namely semidirect products with free groups, determined by a homomorphism from a free group into an appropriate automorphism group such as the mapping class group of a surface. General graphs of surface groups are determined by homomorphisms into the commensurability mapping class group, a much more mysterious object, and this point of view will be used to investigate constructions of new and interesting examples. Also, recent work on graphs of abelian groups has revealed a lot of rich structure, suggesting a real possibility of obtaining QI-rigidity in many new cases. In the second part of this project, the focus will be to study geometric properties of free groups, motivated by analogies between surface groups and free groups. In particular, Thurston's ending lamination conjecture, an important goal in the study of surface groups, has an analogue in the study of free groups, in terms of classifying certain group actions up to equivariant quasi-isometry. Pursuing this issue will require generalizing many of the standard tools of surface groups, such as geodesics in Teichmuller space, to the setting of free groups. Geometric group theory is the study of infinitely symmetric patterns. Popular examples called ``surface groups'' are familiar from wallpaper symmetries and from the symmetries of Escher's prints. Scientific examples occur in the symmetry groups of crystalline arrays, and the symmetry groups of field theories in particle physics. The development of topology starting in the late 19th century, and the concomitant development of combinatorial group theory, exhibited a direct link between abstract groups and geometry. The need for deeper understanding of this link has been demonstrated again and again by different threads within 20th century mathematical developments. Many of these threads were pulled together around 1980 by Gromov, whose proposed unification of geometric group theory using the relation of ``quasi-isometry'' has been very fruitful in the intervening twenty years. The focus of this research project will be to investigate quasi-isometric classification problems for several different types of symmetry groups. In particular, by using a constructive technique known as ``graphs of groups'', new symmetry groups can be constructed out of familiar examples such as surface groups; these and closely related constructions will be the subjects of this research project.