Grant: $96,339 - National Science Foundation - Jun. 17, 2009
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Award Description: 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 residuall 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.
Project Description: In this project, the PI proposed to attack various questions relating to the theory of hyperbolic groups. Motivation for these questions included the question of which hyperbolic groups are residually finite, the question of whether the elementary theory of every hyperbolic group is decidable, and the question of which hyperbolic groups contain surface subgroups. In the short time since the beginning of the grant, there has been progress relating to the first and third questions. (a) The PI has completed his joint work with Owen Cotton-Barratt (University of Oxford) on conjugacy separable hyperbolic groups. As well as a conjugacy separable version of the Rips construction (described in the proposal) numerous other examples of conjugacy separable hyperbolic groups were also found. This has been submitted for publication under the title 'Conjugacy separability of 1-acylindrical graphs of free groups'. (b) There has been further progress on the PI's project with Eric Chesebro and Jason DeBlois. Specifically, the PI and his coauthors have found numerous interesting examples of hyperbolic 3-manifolds to which the main theorem of their paper applies. (c) In joint work with Sang-Hyun Kim, the PI introduced the notion of a polygonal word in a free group. Using this, the PI and Kim have found many new examples of doubles of hyperbolic groups with surface subgroups.
Jobs Summary: A post doc was appointed for a total of .67 FTE. Calculations of Number of Jobs were made using OMB guidance. (Total jobs reported: 1)
Project Status: Less Than 50% Completed
This award's data was last updated on Jun. 17, 2009. Help expand these official descriptions using the wiki below.