Every knot has characterising slopes

Let K be a knot in the 3-sphere. A slope p/q is said to be characterising for K if whenever p/q surgery on K is homeomorphic, via an orientation-preserving homeomorphism, to p/q surgery on another knot K' in the 3-sphere, then K and K' are isotopic. It was an old conjecture of Gordon, proved by Kronheimer, Mrowka, Ozsvath and Szabo, that every slope is characterising for the unknot. In this paper, we show that every knot K has infinitely many characterising slopes, confirming a conjecture of Baker and Motegi. In fact, p/q is characterising for K provided |p| is at most |q| and |q| is sufficiently large.Comment: 15 pages, no figures; final versio

The Heegaard genus of amalgamated 3-manifolds

Let M and M' be simple 3-manifolds, each with connected boundary of genus at least two. Suppose that M and M' are glued via a homeomorphism between their boundaries. Then we show that, provided the gluing homeomorphism is `sufficiently complicated', the Heegaard genus of the amalgamated manifold is completely determined by the Heegaard genus of M and M' and the genus of their common boundary. Here, a homeomorphism is `sufficiently complicated' if it is the composition of a homeomorphism from the boundary of M to some surface S, followed by a sufficiently high power of a pseudo-Anosov on S, followed by a homeomorphism to the boundary of M'. The proof uses the hyperbolic geometry of the amalgamated manifold, generalised Heegaard splittings and minimal surfaces.Comment: 7 pages, 2 figure

Expanders, rank and graphs of groups

Let G be a finitely presented group, and let {G_i} be a collection of finite index normal subgroups that is closed under intersections. Then, we prove that at least one of the following must hold: 1. G_i is an amalgamated free product or HNN extension, for infinitely many i; 2. the Cayley graphs of G/G_i (with respect to a fixed finite set of generators for G) form an expanding family; 3. inf_i (d(G_i)-1)/[G:G_i] = 0, where d(G_i) is the rank of G_i. The proof involves an analysis of the geometry and topology of finite Cayley graphs. Several applications of this result are given.Comment: 13 pages; to appear in Israel J. Mat

Covering spaces of 3-orbifolds

Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime p. This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of O_i grows linearly in the covering degree. The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about p-adic analytic pro-p groups. This has several consequences. Firstly, the fundamental group of O has at least exponential subgroup growth. Secondly, the covers {O_i} have positive Heegaard gradient. Thirdly, we use it to show that a group-theoretic conjecture of Lubotzky and Zelmanov would imply that O has large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3-orbifold with non-empty singular locus has large fundamental group. Many of the above results apply also to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with non-empty singular locus. This includes all closed orientable hyperbolic 3-manifolds with rank two fundamental group, and all arithmetic 3-manifolds.Comment: 26 pages. Version 3 has only minor changes from versions 1 and 2. To appear in Duke Mathematical Journa

New lower bounds on subgroup growth and homology growth

We establish new strong lower bounds on the (subnormal) subgroup growth of a large class of groups. This includes the fundamental groups of all finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic groups. The lower bound is nearly exponential, which should be compared with the fastest possible subgroup growth of any finitely generated group. This is achieved by free non-abelian groups and is slightly faster than exponential. As a consequence, we obtain good estimates on the number of covering spaces of a hyperbolic 3-manifold with given covering degree. We also obtain slightly weaker information on the number of covering spaces of closed 4-manifolds with non-positive Euler characteristic. The results on subgroup growth follow from a new theorem which places lower bounds on the rank of the first homology (with mod p coefficients) of certain subgroups of a group. This is proved using a topological argument.Comment: 39 pages, 2 figures; v3 has minor changes from v2, incorporating referee's comments; v2 has minor changes from v1; to appear in the Proceedings of the London Mathematical Societ

The efficient certification of knottedness and Thurston norm

We show that the problem of determining whether a knot in the 3-sphere is non-trivial lies in NP. This is a consequence of the following more general result. The problem of determining whether the Thurston norm of a second homology class in a compact orientable 3-manifold is equal to a given integer is in NP. As a corollary, the problem of determining the genus of a knot in the 3-sphere is in NP. We also show that the problem of determining whether a compact orientable 3-manifold has incompressible boundary is in NP.Comment: 101 pages, 24 figures; v2 contains some improvements suggested by the referee, which have strengthened the main theorem

Some 3-manifolds and 3-orbifolds with large fundamental group

We provide two new proofs of a theorem of Cooper, Long and Reid which asserts that, apart from an explicit finite list of exceptional manifolds, any compact orientable irreducible 3-manifold with non-empty boundary has large fundamental group. The first proof is direct and topological; the second is group-theoretic. These techniques are then applied to prove a string of results about (possibly closed) 3-orbifolds, which culminate in the following theorem. If K is a knot in a compact orientable 3-manifold M, such that the complement of K admits a finite-volume hyperbolic structure, then the orbifold obtained by assigning a singularity of order n along K has large fundamental group, for infinitely many positive integers n. We also obtain information about this set of values of n. When M is the 3-sphere, this has implications for the cyclic branched covers over the knot. In this case, we may also weaken the hypothesis that the complement of K is hyperbolic to the assumption that K is non-trivial.Comment: 14 pages, 1 figur