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Marc Lackenby

Publications and source records attributed to Marc Lackenby.

At least 55 records · Page 3Linked to original sources
arXiv2008

Spectral geometry, link complements and surgery diagrams

We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require 'complicated' surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds.

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arXiv2008

The maximal number of exceptional Dehn surgeries

We prove two conjectures of C. Gordon. We show that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10, and that the maximal intersection number between exceptional slopes is 8. The proof uses a combination of new geometric techniques and a rigorous computer-assisted calculation.

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arXiv2008

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.

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arXiv2008

Covering spaces of arithmetic 3-orbifolds

Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has infinite virtual first Betti number and G has super-exponential subgroup growth. Our first result, which forms the basis for the entire paper, is that G is commensurable with a lattice containing the Klein group of order 4. Our second result is that if G has a finite index subgroup with first Betti number at least 4, then G is large. This is known to hold for many arithmetic lattices. In particular, we show that it is always the case when G contains A_4, S_4 or A_5. Our third main result is that the Lubotzky-Sarnak conjecture and the geometrisation conjecture together imply that any arithmetic Kleinian group G is large. We also give a new 'elementary' proof that arithmetic Kleinian groups do not have the congruence subgroup property, which avoids the use of the Golod-Shafarevich inequality.

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arXiv2008

An algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary

We provide an algorithm to determine the Heegaard genus of simple 3-manifolds with non-empty boundary. More generally, we supply an algorithm to determine (up to ambient isotopy) all the Heegaard splittings of any given genus for the manifold. As a consequence, the tunnel number of a hyperbolic link is algorithmically computable. Our techniques rely on Rubinstein's work on almost normal surfaces, and also a new structure called a partially flat angled ideal triangulation.

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arXiv2007

Adding high powered relations to large groups

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. The main theorem of the paper is as follows. Let G be a finitely generated, large group and let g_1,...,g_r be a collection of elements of G. Then G/< > is also large, for infinitely many integers n. Furthermore, when G is free, this holds for all but finitely many n. These results have the following application to Dehn surgery. Let M be a compact orientable 3-manifold with boundary a torus. Suppose that the 3-manifold obtained by Dehn filling some slope on the boundary has large fundamental group. Then this is true for infinitely many filling slopes.

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arXiv2007

Detecting large groups

Let G be a finitely presented group, and let p be a prime. Then G is 'large' (respectively, 'p-large') if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a variety of new methods for detecting whether G is large or p-large. These relate to the group's profinite and pro-p completions, to its first L2-Betti number and to the existence of certain finite index subgroups with 'rapid descent'. The paper draws on new topological and geometric techniques, together with a result on error-correcting codes.

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arXiv2006

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.

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arXiv2005

Large groups, Property (tau) and the homology growth of subgroups

We investigate the homology of finite index subgroups G_i of a given finitely presented group G. Specifically, we examine d_p(G_i), which is the dimension of the first homology of G_i, with mod p coefficients. We say that a collection of finite index subgroups {G_i} has linear growth of mod p homology if the infimum of d_p(G_i)/[G:G_i] is positive. We show that if this holds and each G_i is normal in its predecessor and has index a power of p, then one of the following possibilities must be true: G is large (that is, some finite index subgroup admits a surjective homomorphism onto a non-abelian free group) or G has Property (tau) with respect to {G_i}. The arguments are based on the geometry and topology of finite 2-complexes. This has several consequences. It implies that if the pro-p completion of a finitely presented group G has exponential subgroup growth, then G has Property (tau) with respect to some nested sequence of finite index subgroups. It also has applications to low-dimensional topology. We use it to prove that a group-theoretic conjecture of Lubotzky-Zelmanov would imply the following: any lattice in PSL(2,C) with torsion is large. We also relate linear growth of mod p homology to the existence of certain important error-correcting codes: those that are `asymptotically good', which means that they have large rate and large Hamming distance.

math.GR↗
arXiv2005

Heegaard splittings, the virtually Haken conjecture and Property tau

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings. We first give a necessary and sufficient condition, in terms of spectral geometry, for a finitely presented group to have a finite index subgroup with infinite abelianisation. For negatively curved 3-manifolds, we show that this is equivalent to a statement about generalised Heegaard splittings. We also formulate a conjecture about the behaviour of Heegaard genus under finite covers which, together with a conjecture of Lubotzky and Sarnak about Property tau, would imply the virtually Haken conjecture for hyperbolic 3-manifolds. Along the way, we prove a number of unexpected theorems about 3-manifolds. For example, we show that for any closed 3-manifold that fibres over the circle with pseudo-Anosov monodromy, any cyclic cover dual to the fibre of sufficiently large degree has an irreducible, weakly reducible Heegaard splitting. Also, we establish lower and upper bounds on the Heegaard genus of the congruence covers of an arithmetic hyperbolic 3-manifold, which are linear in volume.

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arXiv2005

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.

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arXiv2005

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.

math.GR↗
arXiv2005

A characterisation of large finitely presented groups

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. In this paper, we give a necessary and sufficient condition for a finitely presented group to be large, in terms of the existence of a normal series where successive quotients are finite abelian groups with sufficiently large rank and order. The proof of this result involves an analysis of the geometry and topology of finite Cayley graphs. Theorems of Baumslag and Pride, and their extensions by Gromov and Stohr, on groups with more generators than relations, follow immediately.

math.GR↗
arXiv2003

The asymptotic behaviour of Heegaard genus

Let M be a closed orientable 3-manifold with a negatively curved Riemannian metric. Let {M_i} be a collection of finite regular covers with degree d_i. (1) If the Heegaard genus of M_i grows more slowly than the square root of d_i, then M_i has positive first Betti number for all sufficiently large i. (2) The strong Heegaard genus of M_i cannot grow more slowly than the square root of d_i. (3) If the Heegaard genus of M_i grows more slowly than the fourth root of d_i, then M_i fibres over the circle for all sufficiently large i. These results provide supporting evidence for the Heegaard gradient conjecture and the strong Heegaard gradient conjecture. As a corollary to (3), we give a necessary and sufficient condition for M to be virtually fibred in terms of the Heegaard genus of its finite covers.

math.GT↗
arXiv2003

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.

math.GT↗
arXiv2003

The volume of hyperbolic alternating link complements

If a hyperbolic link has a prime alternating diagram D, then we show that the link complement's volume can be estimated directly from D. We define a very elementary invariant of the diagram D, its twist number t(D), and show that the volume lies between v_3(t(D) - 2)/2 and v_3(16t(D) - 16), where v_3 is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3-manifolds, in the geometric topology. The appendix by Ian Agol and Dylan Thurston, which was written after the first version of this paper was distributed, improves the upper bound on volume to v_3(10t(D) - 10). In addition, examples of alternating links are given which asymptotically achieve this bound.

math.GT↗
arXiv2003

Attaching handlebodies to 3-manifolds

The main theorem of this paper is a generalisation of well known results about Dehn surgery to the case of attaching handlebodies to a simple 3-manifold. The existence of a finite set of `exceptional' curves on the boundary of the 3-manifold is established. Provided none of these curves is attached to the boundary of a disc in a handlebody, the resulting manifold is shown to be word hyperbolic and `hyperbolike'. We then give constructions of gluing maps satisfying this condition. These take the form of an arbitrary gluing map composed with powers of a suitable homeomorphism of the boundary of the handlebodies.

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