Regenerating hyperbolic cone structures from Nil

Let O be a three-dimensional Nil-orbifold, with branching locus a knot Sigma transverse to the Seifert fibration. We prove that O is the limit of hyperbolic cone manifolds with cone angle in (pi-epsilon, pi). We also study the space of Dehn filling parameters of O-Sigma. Surprisingly it is not diffeomorphic to the deformation space constructed from the variety of representations of O-Sigma. As a corollary of this, we find examples of spherical cone manifolds with singular set a knot that are not locally rigid. Those examples have large cone angles.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol6/paper24.abs.htm

Degree one maps between small 3-manifolds and Heegaard genus

We prove a rigidity theorem for degree one maps between small 3-manifolds using Heegaard genus, and provide some applications and connections to Heegaard genus and Dehn surgery problems.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-56.abs.htm

All roots of unity are detected by the A-polynomial

For an arbitrary positive integer n, we construct infinitely many one-cusped hyperbolic 3-manifolds where each manifold's A-polynomial detects every n-th root of unity. This answers a question of Cooper, Culler, Gillet, Long, and Shalen as to which roots of unity arise in this manner.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-11.abs.htm

On the Tits alternative for PD(3)PD(3) groups

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by fewer than four elements always contains a rank 2 free group.Comment: 16 pages, minor corrections, to appear in Ann. Fac. Sci. Toulouse Mat

On invertible generating pairs of fundamental groups of graph manifolds

We study invertible generating pairs of fundamental groups of graph manifolds, that is, pairs of elements (g,h) for which the map g --> g^{-1}, h --> h^{-1} extends to an automorphism. We show in particular that a graph manifold is of Heegaard genus 2 if and only if its fundamental group has an invertible generating pair.Comment: This is the version published by Geometry & Topology Monographs on 29 April 200

The profinite completion of 33-manifold groups, fiberedness and the Thurston norm

We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.Comment: 22 page

On cyclic branched coverings of prime knots

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theorem, a result concerning the symmetries of knots is also obtained. This latter result can be interpreted as a characterisation of the trivial knot.Comment: 28 pages, 2 figure