Geometric Group Theory

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AbstractAward: DMS-0103208Principal Investigator: Lee MosherThe motivating theme of geometric group theory is that finitelygenerated groups can be studied using topological and geometricmethods. Starting around 1980, Gromov proposed classifyinggroups geometrically using the relation of quasi-isometry. Hedemonstrated that the quasi-isometry class of a group G can oftenbe described explicitly in simple algebraic or geometric terms, aprocess now referred to as ``quasi-isometric rigidity'' or``QI-rigidity'' for the group G. The first part of this projectwill be to investigate QI-rigidity problems for graphs of surfacegroups and of abelian groups. An initial focus will be thosegraphs of groups with the simplest algebraic structure, namelysemidirect products with free groups, determined by ahomomorphism from a free group into an appropriate automorphismgroup such as the mapping class group of a surface. Generalgraphs of surface groups are determined by homomorphisms into thecommensurability mapping class group, a much more mysteriousobject, and this point of view will be used to investigateconstructions of new and interesting examples. Also, recent workon graphs of abelian groups has revealed a lot of rich structure,suggesting a real possibility of obtaining QI-rigidity in manynew cases. In the second part of this project, the focus will beto study geometric properties of free groups, motivated byanalogies between surface groups and free groups. In particular,Thurston's ending lamination conjecture, an important goal in thestudy of surface groups, has an analogue in the study of freegroups, in terms of classifying certain group actions up toequivariant quasi-isometry. Pursuing this issue will requiregeneralizing many of the standard tools of surface groups, suchas geodesics in Teichmuller space, to the setting of free groups.Geometric group theory is the study of infinitely symmetricpatterns. Popular examples called ``surface groups'' arefamiliar from wallpaper symmetries and from the symmetries ofEscher's prints. Scientific examples occur in the symmetrygroups of crystalline arrays, and the symmetry groups of fieldtheories in particle physics. The development of topologystarting in the late 19th century, and the concomitantdevelopment of combinatorial group theory, exhibited a directlink between abstract groups and geometry. The need for deeperunderstanding of this link has been demonstrated again and againby different threads within 20th century mathematicaldevelopments. Many of these threads were pulled together around1980 by Gromov, whose proposed unification of geometric grouptheory using the relation of ``quasi-isometry'' has been veryfruitful in the intervening twenty years. The focus of thisresearch project will be to investigate quasi-isometricclassification problems for several different types of symmetrygroups. In particular, by using a constructive technique knownas ``graphs of groups'', new symmetry groups can be constructedout of familiar examples such as surface groups; these andclosely related constructions will be the subjects of thisresearch project.