Principal Investigator: Genevieve S. Walsh
Two n-manifolds are commensurable if they have homeomorphic finite-sheeted covers. This notion is particularly useful for 3-manifolds. For 3-manifolds which admit a geometry, commensurability respects this geometry. In light of the recent proof of geometrization for 3-manifolds, classification into commensurability classes is a refinement of the classification of 3-manifolds by which geometry they admit. This applies to 3-manifolds which have been decomposed along 2-spheres and 2-tori. Questions concerning virtual properties of $3$-manifolds, such as if a 3-manifold is virtually Haken or virtually fibered, are also questions about its commensurability class. Commensurability also respects certain number-theoretic properties of 3-manifolds. A classification of hyperbolic 3-manifolds up to commensurability would be very useful for the field. A first step towards understanding commensurability classes of hyperbolic 3-manifolds is understanding commensurability classes of hyperbolic knot complements. The PI proposes to work on the conjecture that there are at most 3 hyperbolic knot complements in a given commensurability class, and to understand how these commensurabilities can occur. Commensurability is also a useful equivalence relation on hyperbolic surfaces, with the definition that two hyperbolic 2-manifolds are commensurable if they have isometric finite-sheeted covers. The PI proposes to further develop this theory, in particular to understand commensurability classes of surfaces with large symmetry groups.
3-manifolds are spaces which locally look like a 3-dimensional ball. In particular, our universe is a 3-manifold and we do not know which one it is. A deeper understanding of commensurabilities amongst 3-manifolds would significantly refine geometrization for 3-manifolds. Knot complements are a naturally occurring class of 3-manifolds, and thus good candidates for study. 2-manifolds, or surfaces, are spaces which locally look like a 2-dimensional disc. Hyperbolic surfaces are the most common type of surface and are used in an extremely wide variety of scientific contexts. The study of symmetry and commensurability is very useful for these surfaces. The PI regularly speaks on her work to diverse audiences. Any results from this research will be described in research seminars, posted on the arXiv, and broadly distributed.
