Essential closed surfaces in bounded 3-manifolds

Suppose M is a connected, open, orientable, irreducible 3-manifold which is not homeomorphic to R^3. Given a compact 3-manifold J in M which satisfies certain conditions, Brin and Thickstun have associated to it an open neighborhood V$ called an end reduction of M at J. It has some useful properties which allow one to extend to M various results known to hold for the more restrictive class of eventually end irreducible open 3-manifolds. In this paper we explore the relationship of V and M with regard to their fundamental groups and their covering spaces. In particular we give conditions under which the inclusion induced homomorphism on fundamental groups is an isomorphism. We also show that if M has universal covering space homeomorphic to R^3, then so does V. This work was motivated by a conjecture of Freedman (later disproved by Freedman and Gabai) on knots in M which are covered by a standard set of lines in R^3.

Chapter 26 - Wall's Finiteness Obstruction

Isotopies of 3-manifolds

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.

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 .

Groups associated to a knot in S3 which classify the knot have been constructed by Conway and Gordon [4], Simon [12] and Whitten[16]; see also [1] and [19] for certain special classes of knots. In this note we show that for a knot K in S3 of the groups π1(S3/K)/〈malb〉 classify the knot; here m resp. l denote a meridian resp. longitude of the knot and 〈〉 denotes normal closure. This is based on Thurston's orbifold geometrization theorem [13–15] and the following result (see also [8, 19]):Theorem 1. Let O1 and O2 be good closed irreducible 3-orbifolds which possess a decomposition into geometric pieces (along Euclidean 2-suborbifolds, see [2, 15]) and have infinite (orbifold-) fundamental group. Then O1 and O2 are diffeomorphic (as orbifolds) if and only if π1 O1 and π1 02 are isomorphic.

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 [6] it is shown that two (compact) orientable 3-manifolds which are irreducible, boundary irreducible and sufficiently large are homeomorphic if and only if there exists an isomorphism between the fundamental groups which respects the peripheral structure. In this note we extend this theorem to reducible 3-manifolds.Any compact 3-manifold M has a decomposition into prime manifolds [1; 4].1Here the connected sum of two bounded manifolds N1, N2 is denned by removing 3-balls B1 B2 in int N1, int N2, respectively, and glueing the resulting boundary spheres together. The M1's which occur in the decomposition (1) are either irreducible or handles (i.e., a fibre bundle over S1 with fibre S2). If (1) contains a fake 3-sphere, we assume it to be Mn.

The fundamental group at infinity

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 .

Cartesian powers of 3-manifolds

Ribbonness of a stable-ribbon surface-link, I. A stably trivial surface-link

This meeting was focused on asymptotic aspects of group theory. The resulting problems lead, in particular, to the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients. Also, properties of infinite families of finite groups are at the center of interest in the field. We had talks on the most important recent developments in the field of asymptotic and profinite group theory. We single out some of these new results. The classification of finite simple groups has provided an impetus for the detailed investigation of properties of families of groups of Lie type and of alternating groups. This has produced many results of asymptotic or probabilistic nature, describing in a quantitative way properties that hold for all sufficiently large simple groups, or else that hold ‘almost surely’ as the orders of the groups are tending to infinity. Here we had a talk of Aner Shalev on his recent solution (with coauthors) of Ore's conjecture. Recently there has been much activity in the study of infinite groups, with an emphasis on the asymptotic behaviour of their finite quotients, or their finite-index subgroups. The theory of subgroup growth is now well established; the theory of the associated zeta functions is making good progress. More subtle variations of subgroup growth are also developing into significant theories in their own right: there are important new results on representation growth – of arithmetic groups, nilpotent groups and pro-p groups; and on maximal subgroup growth, with applications to probabilistic generation properties. Profinite groups are ‘asymptotic limits’ of finite groups. The nature of verbal subgroups and verbal mappings, in both finite and profinite groups, has been the focus of much recent work. This has led among other things to the solution of Serre's problem on subgroups of finite index in finitely generated profinite groups; other significant results include the characterisation of closed verbal subgroups in pro-p groups and new characterisations of solubility in finite groups. Branch groups, both discrete and profinite, are certain groups of automorphisms of rooted trees; introduced by Grigorchuk and others over 20 years ago, they have provided the solution to several outstanding problems, and their study continues to provide new results. Some of them were reported on in our meeting. We also had talks from important fields of mathematics in which profinite and asymptotic group theory finds application. There is, for instance, the study of the absolute Galois group of the rationals and in particular its closed pro-p subgroups. Methods from pro-p group theory have successfully been applied in order to understand fundamental groups of (hyperbolic) 3-manifolds. The interplay between the topological fundamental group of a complex algebraic surface and its profinite completion which is the etale fundamental group of the surface is the subject of recent research. We believe that the meeting was exciting and successful. The high quality lectures covered a lot of very recent results in the field. There were many intense discussions between mathematicians of different fields. Altogether there were over 50 participants from all over the world. The percentage of young researchers was very high. The following extended abstracts were collected and edited by Dr. Evija Ribnere.