Separability and logic in geometric group theory

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AbstractAward: DMS-0906276Principal Investigator: Henry WiltonThis award is funded under the American Recovery and ReinvestmentAct of 2009 (Public Law 111-5).The principal investigator will pursue several lines of researchmotivated by three deep open questions in the theory ofword-hyperbolic groups. (1) Is every hyperbolic group residuallyfinite? If so, then in fact word-hyperbolic groups satisfy muchstronger separability properties. The PI intends to investigatethese separability properties for known examples ofword-hyperbolic groups and related groups, including 3-manifoldgroups and relatively hyperbolic groups. (2) Is the elementarytheory of a torsion-free hyperbolic group decidable? Togetherwith Daniel Groves, the PI intends to give an affirmative answerby proving algorithmic versions of the results of Sela. (3) Doesevery word-hyperbolic group contain a surface subgroup? Little isknown about this famous question of Gromov, even for some verybasic examples of word-hyperbolic groups. The PI proposes to usevarious new techniques to find surface subgroups in examples ofword-hyperbolic groups, including doubles of free groups alongmaximal cyclic subgroups.A group is a collection of symmetries, for example the collectionof all translations of the Euclidean plane that preserve theinteger lattice. This group is generated by the translations ofone unit of length in the north, south, east, or west directions,in the sense that any element of the group can be obtained bycomposing copies of those four basic translations. Everyfinitely generated group carries a notion of distance betweenpairs of its elements, defined by counting the number ofgenerators that must be applied to carry one element of the groupto another. These projects concentrate on word-hyperbolicgroups, which have many of the properties of the hyperbolic planefrom non-Euclidean geometry and are known to be so common that arandomly selected finitely generated group is almost surelyword-hyperbolic. This award is jointly funded by the programs inTopology and Foundations.