We prove that an irreducible orientable 3-manifold M of euler characteristic zero which is a subgroup of a right-angled reflection group is virtually fibered. As a consequence, arithmetic hyperbolic 3-manifolds defined by quadratic forms and finite-volume reflection 3-orbifolds virtually fiber. We will discuss a more general group-theoretic criterion for fibering called RFRS, and formulate a program involving a series of conjectures which would imply that hyperbolic 3-manifolds virtually fiber.
Lorenz knots and links are the periodic orbits in a flow on R3 that was discovered by E.N. Lorenz in 1963, and has since become a paradigm for chaos. In our talk we will discuss Lorenz knots and links from three viewpoints. The first is from the viewpoint of dynamical systems. The second relates to hyperbolic geometry, and is joint work with Ilya Kofman. The third relates to the modular group, and a discovery of Etienne Ghys. This unusual family of links appears to play a more basic role in mathematics than anyone had suspected.
Sharp bounds have been determined for the distance between most kinds of exceptional filling slopes on the boundary of hyperbolic knot manifolds. The one family of slopes yet to be successfully dealt with are those which yield small Seifert spaces. In this talk I will report on joint works with Marc Culler, Cameron Gordon, Peter Shalen, and Xingru Zhang where we obtain various optimal bounds, and close-to-optimal bounds, for the distance between a reducible filling slope and a small Seifert filling slope.
We investigate the analog of the Thurston boundary of Teichmuller space in the context of convex real projective structures on closed manifolds. In particular we give a new interpretation of measured laminations in terms of non-standard hyperbolic structures over the hyper-reals.
Let S be an orientable surface of finite type, and Mod(S) its mapping class group.
If B is a compact space, and G is its fundamental group then the
set of isomorphism classes of S-bundles over B is naturally
parametrized by:
I'll present some results about the structure of the set XS(G), for an arbitrary finitely presented group G. Morally, XS(G) being infinite implies that G admits a splitting as a graph of groups, and the structure of the set XS(G) can (often) be understood via studying splittings of G.
We discuss the AJ conjecture that relates the Jones polynomial to the A-polynomial of knots.
We show that if two 3-manifolds with connected boundary are glued together using a map of high complexity, then the low-genus Heegaard splittings of the resulting closed 3-manifold are standard in a certain sense. The complexity of the gluing map is defined using curve complex and the boundary curves of certain properly embedded surfaces in the two 3-manifolds.