Principal Investigator: Henry Wilton
This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The principal investigator will pursue several lines of research motivated by three deep open questions in the theory of word-hyperbolic groups. (1) Is every hyperbolic group residually finite? If so, then in fact word-hyperbolic groups satisfy much stronger separability properties. The PI intends to investigate these separability properties for known examples of word-hyperbolic groups and related groups, including 3-manifold groups and relatively hyperbolic groups. (2) Is the elementary theory of a torsion-free hyperbolic group decidable? Together with Daniel Groves, the PI intends to give an affirmative answer by proving algorithmic versions of the results of Sela. (3) Does every word-hyperbolic group contain a surface subgroup? Little is known about this famous question of Gromov, even for some very basic examples of word-hyperbolic groups. The PI proposes to use various new techniques to find surface subgroups in examples of word-hyperbolic groups, including doubles of free groups along maximal cyclic subgroups.
A group is a collection of symmetries, for example the collection of all translations of the Euclidean plane that preserve the integer lattice. This group is generated by the translations of one unit of length in the north, south, east, or west directions, in the sense that any element of the group can be obtained by composing copies of those four basic translations. Every finitely generated group carries a notion of distance between pairs of its elements, defined by counting the number of generators that must be applied to carry one element of the group to another. These projects concentrate on word-hyperbolic groups, which have many of the properties of the hyperbolic plane from non-Euclidean geometry and are known to be so common that a randomly selected finitely generated group is almost surely word-hyperbolic. This award is jointly funded by the programs in Topology and Foundations.
