This paper stems from the observation (arising from work of Delzant) that “most” Kähler groups $G$ virtually algebraically fiber, that is, admit a finite index subgroup that maps onto $\mathbb{Z}$ with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, that is, they have virtual Albanese dimension $va(G)\leqslant 1$ . We show that the existence of algebraic fibrations has implications in the study of coherence and higher BNSR invariants of the fundamental group of aspherical Kähler surfaces. The class of Kähler groups with $va(G)=1$ includes virtual surface groups. Further examples exist; nonetheless, they exhibit a strong relation with surface groups. In fact, we show that the Green–Lazarsfeld sets of groups with $va(G)=1$ (virtually) coincide with those of surface groups, and furthermore that the only virtually RFRS groups with $va(G)=1$ are virtually surface groups.

It follows from earlier work of Silver and Williams and the authors that twisted Alexander polynomials detect the unknot and the Hopf link. We now show that twisted Alexander polynomials also detect the trefoil and the figure-8 knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split, and whether it is totally split.

The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the splitting numbers of links with nine or fewer crossings. Also, with these techniques, we either reprove or improve upon the lower bounds for splitting numbers of links computed by Batson and Seed using Khovanov homology.

In [3] the authors (M. Borodzik and S. Friedl, Unknotting number and classical invariants (preprint 2012)) associated to a knot K ⊂ S3 an invariant nℝ(K), which is defined using the Blanchfield form and which gives a lower bound on the unknotting number. In this paper, we express nℝ(K) in terms of the Levine-Tristram signatures and nullities of K. We also show in the proof that the Blanchfield form for any knot K is diagonalisable over ℝ[t±1].

We give new Casson–Gordon style obstructions for a two–component link to be topologically concordant to the Hopf link.

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

Let M be a closed 4-manifold with a free circle action. If the orbit manifold N3 satisfies an appropriate fibering condition, then we show how to represent a cone in H2(M; ℝ) by symplectic forms. This generalizes earlier constructions by Thurston, Bouyakoub and Fernández et al. In the case that M is the product 4-manifold S1 × N, our construction complements our previous results and allows us to determine completely the symplectic cone of such 4-manifolds.

Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

We study eta-invariants of links and show that in many cases they form link concordance invariants, in particular that many eta-invariants vanish for slice links. This result contains and generalizes previous invariants by Smolinsky and Cha–Ko. We give a formula for the eta-invariant for boundary links. In several interesting cases this allows us to show that a given link is not slice. We show that even more eta-invariants have to vanish for boundary slice links.

Cochran, Orr and Teichner introduced $L^2$-eta-invariants to detect highly non-trivial examples of non slice knots. Using a recent theorem by Lück and Schick we show that their metabelian $L^2$-eta-invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction theorem proved by the author using finite dimensional unitary eta-invariants. We show that if for a knot $K$ this ribbon obstruction vanishes then the metabelian $L^2$-eta-invariant vanishes too. The converse has been shown by the author not to be true.