The PI will work with Behrstock to complete the quasi-isometric classification of 3-manifold groups (or equivalently, the bi-Lipschitz classification of universal covers of 3-manifolds), which has involved a large number of researchers for over a decade. He will also use bi-Lipschitz geometry and knot-theoretic tools in the search for more flexible approaches to classification questions for complex surface singularities. For hyperbolic manifolds there are postulated connections between geometric, representation-theoretic, and quantum based invariants of manifolds, and the PI will work towards specific conjectures relating these invariants. The PI will also continue his investigation of commensurability properties of 3-manifolds.
This proposal addresses questions concerning 3-dimensional space forms that arise from disparate areas of mathematics. These questions create interactions of low dimensional topology with algebra, algebraic geometry, number theory, and theoretical physics. Links between disparate areas provide much of the power of mathematics, and strengthening these links increases the power. Specific tools that will be used are the study of the geometry up to bounded distortion ("bi-Lipschitz geometry") and the use of algebraic and number-theoretic invariants.
