Some groups which classify knots

Chapter 26 - Wall's Finiteness Obstruction

Isotopies of 3-manifolds

An irreducible open 3-manifold W is R 2 -irreducible if it contains no non-trivial planes, i.e. given any proper embedded planein W some component of W −� must have closure an embedded halfspace R 2 ×(0, ∞). In this paper it is shown that if M is a connected, P 2 -irreducible, open 3-manifold such that �1(M) is finitely generated and the universal covering space f M of M is R 2 -irreducible, then either f M is homeomorphic to R 3 or �1(M) is a free product of infinite cyclic groups and fundamental groups of closed, connected surfaces other than S 2 or P 2 . Given any finitely generated group G of this form, uncountably many P 2 -irreducible, open 3-manifolds M are constructed with �1(M) ∼ G such that the universal covering space f M is R 2 -irreducible and not homeomorphic to R 3 ; the f M are pairwise non- homeomorphic. Relations are established between these results and the conjecture that the universal covering space of any irreducible, orientable, closed 3-manifold with infinite fundamental group must be homeomorphic to R 3 .

A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i: S 9<M of a closed, orientable connected surface S is essential if it is injective at the level of fundamental groups and the group i*rr1 (S) cannot be conjugated into a subgroup 7rr(coM) of -rr(M), where &oM is a component of OM. This latter condition is equivalent to the statement that the image of the surface S cannot be freely homotoped into OM. One of the main results of this paper is the following:

We study the existence of pluricanonical forms on an algebraic variety of general type with infinite fundamental group. We have a nonvanishing of m th plurigenus of the variety by that of a general fibre of the so-called Shafarevich map of the variety.

Let M be an oriented irreducible 3-manifold with infinite fundamental group and empty or toroidal boundary which is not S^1 \times D^2 . Consider any element \phi in the first cohomology of M with integer coefficients. Then one can define the \phi -twisted L^2 -torsion function of the universal covering which is a function from the set of positive real numbers to the set of real numbers. By earlier work of the second author and Schick the evaluation at t=1 determines the volume. In this paper we show that the degree of the L^2 -torsion function, which is a number extracted from its asymptotic behavior at 0 and at \infty , agrees with the Thurston norm of \phi .

We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics.

We introduce and study trace fields and invariant trace fields of SL2(C) and PSL2(C) representations of 3-manifold groups. We give conditions on such fields which imply that the underlying 3-manifold is virtually Haken or has virtually positive or 1 first Betti number. In particular, we define the notion of an algebraically trace-proper surface subgroup in a 3-manifold group, and show that any closed orientable irreducible 3-manifold with such a surface subgroup is virtually Haken. We give infinitely many families of closed orientable hyperbolic non-Haken 3-manifolds with algebraically trace-proper surface subgroups. In this paper a 3-manifold is always assumed to be connected and orientable, without loss of generality for the problems to be considered. A 3-manifold is said to be Haken if it is compact, irreducible, and contains a properly embedded incompressible surface. A 3-manifold is said to be virtually Haken if it has a finite cover which is Haken. The well-known virtually Haken conjecture states that every closed and irreducible 3-manifold with infinite fundamental group is virtually Haken. It is also conjectured that every closed Haken 3-manifold has virtually positive first Betti number, that is, the manifold has a finite cover which has positive first Betti number. If M is a closed hyperbolic Haken 3-manifold, then it is further conjectured that M has virtually 1 first Betti number, which means that for any positive integer n, there is a finite cover Mn of M such that the first Betti number of Mn is larger than n. These conjectures are fundamental and difficult issues in 3-manifold topology. In this paper we provide some information about these conjectures by studying traces of representations of 3-manifold groups into SL2ðCÞ or PSL2ðCÞ: For a group G and a representation r : G ! SL2ðCÞ; we define the trace field Kr of r to be the field generated by the traces of all the matrices r (g), g [ G; over the base field Q of rational numbers, that is, Kr ¼ Qðtrðr ðgÞÞ; g [ G Þ:

In this paper, in an attempt to extend an earlier work of Lück, we construct a knot invariant with parameter in C* by using the fundamental L2-representation of the fundamental group of the knot complement, which may be thought of as an L2-analogue of the usual Alexander polynomial of the knot in S3. When restricted to U(1) parameters, we interpret this invariant in terms of the U(1) twisted L2-Reidemeister torsion. We also show that this L2-invariant depends only on the norm |t| for t ∈ C*. In particular, this implies an unexpected rigidity property of the U(1) twisted L2-torsion on a knot complement. A possible relationship with the volume conjecture is discussed.

Let be a torus embedded in *S and X its exterior. The peripheral subgroup of is the image of nt(dX) by the map /„ induced by the inclusion i: dXc+X. So it is isomorphic to the direct sum of the infinite cyclic group Z, which is generated by a meridian, and some quotient of π1(F) = Z®Z. We denote by τF the second summand and call it the type of F see [10, §3]. If is unknotted, that is, bounds a solid torus in S 4 (see [7]), then τ = 0 . If is a torus constructed by spinning a non-trivial classical knot, then τF=Z, cf. [3, 10, 11]. Asano [1] and Litherland [10] constructed examples with τ = Z φ Z . And Boyle [3] showed that there are tori of type Zn for n = 2,5 and 10 by attaching a 2-dimensional 1-handle to the 5-twist-spun trefoil. We abbreviate the group π^S^ — F) of as πF. It is known that the second homology of the group H2(nF) is a quotient of H2(S ' — F) = Z φ Z, and several authors gave examples having non-trivial second homology [2, 4], see also [12]. Litherland [10] showed that H2(πF) is a quotient of τF and that any quotient of Z φ Z is realizable as H2{πF) for some torus of type Z φ Z . For abelian groups A and JB, we write A <B if A is a quotient of B. It is natural to ask

Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of classical knot theory. Recent works have established Alexander and Markov theorems in the planar setting. In the classical case, the role of groups is played by twin groups, a class of right-angled Coxeter groups. A new class of groups called virtual twin groups, that extends twin groups in a natural way, plays the role of groups in the virtual case. The virtual twin group V Tn contains the pure virtual twin group P V Tn, a planar analogue of the pure Artin braid group. In this paper, we prove that the pure virtual twin group P V Tn is an irreducible right-angled Artin group with trivial center and give it's precise presentation. We show that P V Tn has a decomposition as an iterated semi-direct product of infinite rank free groups. We give a complete description of the automorphism group of P V Tn and establish splitting of natural exact sequences of automorphism groups. As applications, we show that V Tn is residually finite and P V Tn has the R∞-property.

The fundamental group at infinity

We show how the Conway Alexander polynomial arises from theq deformation of (Z 2 graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into theU q sl(1, 1) R matrix by going to the exterior power algebra. Using a det=str identity, this allows us to recover the state model of [K2, 89] as well. We also show how theU q> sl(1, 1) algebra describes free fermions “propagating” on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.

The classical knot groups are the fundamental groups of the complements of smooth or piecewise-linear (PL) locally-flat knots. For PL knots that are not locally-flat, there is a pair of interesting groups to study: the fundamental group of the knot complement and that of the complement of the "boundary knot" that occurs around the singular set, the set of points at which the embedding is not locally-flat. If a knot has only point singularities, this is equivalent to studying the groups of a PL locally-flat disk knot and its boundary sphere knot; in this case, we obtain a complete classification of all such group pairs in dimension ≥6. For more general knots, we also obtain complete classifications of these group pairs under certain restrictions on the singularities. Finally, we use spinning constructions to realize further examples of boundary knot groups.

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and \operatorname{CAT}(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete \mathcal{VH} complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. For the class of complete directed \mathcal{VH} complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise. Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete \mathcal{VH} complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group F_2 .