Programme April 19-21 avril: 10:30-12:00 and/et 14:00-15:30 The structure of groups with a quasiconvex hierarchy We prove that (certain relatively) hyperbolic groups with a quasiconvex hierarchy are virtually subgroups of graph groups.
Our focus is on "special cube complexes" which are nonpositively curved cube complexes that behave like "high dimensional graphs" and are closely related to graph groups. The main result illuminates the structure of a group by showing that it is "virtually special", and this yields the separability of the quasiconvex subgroups of the groups we study.
As an application, we resolve Baumslag's conjecture on the residual finiteness of one-relator groups with torsion. Another application shows that generic haken hyperbolic 3-manifolds have "virtually special" fundamental group. Since graph groups are residually torsion-free nilpotent, combined with Agol's virtual fibering criterion, this proves that finite volume haken hyperbolic 3-manifolds are virtually fibered. April 22 avril9:30-10:30:Essential Immersed Surfaces in Closed Hyperbolic 3-manifolds Given any closed hyperbolic 3-manifold M and e > 0, we find a closed hyperbolic surface S and an immersion f: S → M such that f lifts to a e-almost-isometry from the universal cover of S to the universal cover of M. In particular f induces an injection on the fundamental group of S; thus there is an essential immersed surface in every closed hyperbolic 3-manifold. I will explain why the mixing of the frame flow on M implies the existence of a highly symmetric collection of pairs of pants, which can then be assembled to form the desired surfaces S. 11:00-12:00:Incompressible surfaces in hyperbolic link manifolds We show that every hyperbolic link manifold contains closed quasi-Fuchsian surfaces. This is a joint work with Joseph Masters. 14:00-15:00:Sub-logarithmic Heegaard Gradients The following conjecture of Thurston is still open: every hyperbolic, connected, complete, of finite volume and orientable 3-manifold M admits a finite cover which fibers over the circle. Related to this conjecture, one can consider a new invariant defined by Lackenby, the Heegaard gradient of the manifold M. Lackenby conjectures that this gradient is zero if and only if M has a finite cover fibered over the circle. Building on a result of Maher, we propose a sub-logarithmic variation of this Heegaard gradient and prove Lackenby's conjecture for this sub-logarithmic gradient. This result gives also a criterion to determine if a family of finite covers of M contains a cover in which there is an embedded surface that is a virtual fiber. 15:30-16:30:Twisted Alexander polynomials and the genus of hyperbolic knots Twisted Alexander polynomials give lower bounds on the genus of a knot. We will discuss the question when do twisted Alexander polynomials determine the genus of a hyperbolic knot, and we will discuss the properties of the twisted Alexander polynomial of a hyperbolic knot corresponding to the canonical SL(2,C) representation. April 23 avril9:30-10:30:Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds I will exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many fundamentally distinct ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration of M over the circle is the modular elliptic curve E=X_0(49), which admits multiplication by the ring of integers of Q[sqrt(-7)]. This is joint work with Dinakar Ramakrishnan. 11:00-12:00:Cubulating malnormal amalgams We say that a group G is cubulated if it acts properly and cocompactly (or more generally, cosparsely) on a CAT(0) cube complex. We prove a combination theorem for cubulated groups, a special case of which says (roughly) that if G splits as G=A *_{C} B with quasiconvex vertex and edge groups, A and B are cubulated and hyperbolic relative to virtually abelian subgroups, C is malnormal in A and B, and a few other technical conditions hold, then G is cubulated. 14:00-15:00:Grothendieck's Problem for 3-Manifold Groups The following problem was posed by Grothendieck: Let G be a finitely presented residually finite group and u : H ->G the inclusion homomorphism of a finitely presented subgroup. Suppose that the extension to the profinite completions is an isomorphism. Is u an isomorphism? This talk will discuss this problem in the context of the fundamental groups of compact 3-manifolds. 15:30-16:30:Survey on the virtual Haken problem |