Groups acting on Asymptotically CAT(0) spaces

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Geometric group theory is the mathematical tool for studying symmetry. This is achieved by investigating groups via their actions on geometric spaces. A fundamental class of objects studied by geometric group theorists are manifolds or varieties which are solutions sets to collections of equations. A collection of equations or parameters could arise naturally in different areas of mathematics, of science and in economics. Hence research in geometric group theory is of paramount importance. In this proposal we are interested in groups acting on trees, a theory that was initiated by Bass and Serre and has profound influence on the splittings of groups as amalgamated free products and HNN extensions. If a group splits, then one can decompose it into smaller, more manageable pieces. The fundamental group links group theory to geometry and topology. Splitting the fundamental group of a manifold holds the key in some cases to splitting the manifold itself into smaller better-understood sub-manifolds. This is related to important conjectures in mathematics like the virtual Haken conjecture. We hope that the work arising from this proposal will enlighten us on the phenomenon of splittings of groups and manifolds. A second theme of the project is to exhibit the richness of the class of what we call asymptotically CAT(0) spaces which heuristically are metric spaces appearing to have non-positive curvature when viewed from increasingly distant observation points. Trees and hyperbolic spaces are natural examples. We hope to construct examples of asymptotically CAT(0) graphs which not hyperbolic in the sense of Gromov. This is related to a question of Erdos held in esteem among computer scientists and graph theorists. Evidently a resolution of Erdos' question will allow us to solve geometric problems algorithmically on computer.