Geometry of Groups and Splittings

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Geometry of Groups and SplittingsThis project has four components. The first component entails the studyof higher dimensional filling invariants of groups, with emphasis oncontructing examples exhibiting the full range of possiblebehavior. Forester and collaborators Brady, Bridson, and Shankar intendto develop general methods for computing such invariants, and to studythe role of subgroup distortion and curvature. The second componentconcerns the construction of geometric structures for particular classesof groups, such as limit groups, and the study of the structure ofsubgroups and ends for these groups. In the third component Forester willinvestigate the classification of generalized Baumslag-Solitar groupsand pursue applications. In the fourth component Forester will continuehis work with Rourke on the multivariable adjunction problem, in the caseof torsion-free groups. This work extends and devolops Klyachko'smethods.Group theory is the theory of symmetry, which plays a fundamental role inmathematics. Notions of symmetry (or "groups") arise very naturally ingeometry, but also in other, more abstract, areas. In the field ofgeometric group theory one studies abstract groups by constructinggeometric spaces, tailor-made for the groups at hand, in order to "see"the symmetry and understand it. This project involves the detailed studyof the geometric spaces thus constructed.