The mathematical study of 3-dimensional spaces goes back to the work of Poincar(e. Because classical physics describes our universe to be a 3-dimensional space, the classification of 3-dimensional spaces is an important mathematical endeavor, since it may have ramifications for the global structure of our universe. One of the most broad and interesting classes of 3-dimensional spaces are "hyperbolic manifolds", which are described by discrete groups of two by two complex matrices. These groups can have manifestations as higher-dimensional matrices, which the PI plans to explore. The importance for the study of 3-dimensional spaces may stem from the study of "bundles" over the spaces, which attach vector data to each point in the 3-dimensional space. The PI plans to investigate the moduli of such bundles, both in terms of dimension, and number of components (which structures can be deformed to each other). The project plans to take advantage of recent advances in the understanding of 3-dimensional hyperbolic spaces discovered by the PI and others. There may be ramifications for the algorithmic classification of 3-dimensional spaces and their invariants.
This project proposes several avenues of study regarding representations of 3-manifold groups. The PI will investigate the profinite completions of hyperbolic 3-manifold groups. In particular, the PI will explore which information about a 3-manifold may be extracted from the finite quotients of its fundamental group, including hyperbolic volume and rank. The PI proposes to investigate the dimension of the space of faithful irreducible representations as the dimension of the representations increases. The PI plans to use right-angled Artin groups and embeddings of 3-manifolds into them to investigate this question. A related question is to investigate the growth of the number of volumes of n-dimensional representations of hyperbolic 3-manifold groups as n increases. Finally, the PI will investigate twisted-homology of hyperbolic taut sutured 3-manifolds. In particular, the PI would like to find criteria for when they are twisted SL(2,C)-homology products, motivated by the problem of determining when twisted SL(2,C) Alexander polynomials detect the Seifert genus (or more generally Thurston norm).