By Mostow rigidity, the volume and Chern-Simons invariant of a hyperbolic 3-manifold are topological invariants. They are often regarded as the real and imaginary part of a complex valued invariant known as the complex volume. The complex volume can be defined more generally as an invariant of a parabolic representation of a 3-manifold group into a simple complex Lie group, e.g. SL(n,C). The PI and his collaborators will study such representations and extend several results previously only known for SL(2,C). In particular, we will give a concrete formula for the complex volume. The new idea is that a parabolic representation has a fundamental class in Neumann's extended Bloch group. By a result of the PI, the extended Bloch group can be defined over a number field, and is isomorphic to an algebraic K-group. This may shed new light on the role of the complex volume in number theory.
The volume and Chern-Simons invariant are interesting and important invariants that appear in a wide range of areas of mathematics including mathematical physics, hyperbolic geometry and number theory. The proposal introduces new techniques for computing these invariants via an object known as the extended Bloch group. The extended Bloch group is isomorphic to a group appearing in a different branch of mathematics known as algebraic K-theory. The relationship with K-theory may lead to an improved understanding of the role of the volume and Chern-Simons invariant in number theory, which has so far been completely mysterious.
The volume and Chern-Simons invariant show up naturally in a wide range of areas of mathematics including mathematical physics, hyperbolic geometry, and number theory. The purpose of the project was to study these invariant from a wide range of perspective and develop new ways of computing them. The new techniques have open up new venues for future research. The main outcomes are summarized below. - Development of the first known practical algorithm for exact computations. Computations that were previously infeasible even for simple manifolds are now possible even for very complex ones. - Improved understanding of the volume and Chern-Simons invariants, and their role in number theory and mathematical physics. - Development of software and databases of computations. The software tools have been integrated in the popular open source program SnapPy for computing invariants of hyperbolic manifolds. The results above have been disseminated in the following way: - Publications in peer reviewed journals and on the arXiv (1 publication, 2 preprints). - Development of a database of representations and their volume and Chern-Simons invariant publically available at http://ptolemy.unhyperbolic.org/ - Research presentations at 7 conferences, 16 department seminars and a colloquium. The project has contributed to training and education in the following way. - The PI was a postdoctoral supervisor for Matthias Goerner in spring 2012. - The PI gave minicourses in Korea (Oct 2010) and Denmark (Aug 2012) for graduate students and young researchers.
