We construct a new spectral sequence beginning at the Khovanov homology of a link and converging to the Khovanov homology of the disjoint union of its components. The page at which the sequence collapses gives a lower bound on the splitting number of the link, the minimum number of times its components must be passed through one another in order to completely separate them. In addition, we build on work of Kronheimer and Mrowka and Hedden and Ni to show that Khovanov homology detects the unlink.

Let $G=SL(2,\mathbb{R})⋉{\mathbb{R}}^2$ be the affine special linear group of the plane, and set $\Gamma =SL(2,\mathbb{Z})⋉{\mathbb{Z}}^2$. We prove a polynomially effective asymptotic equidistribution result for the orbits of a $1$-dimensional, nonhorospherical unipotent flow on $\Gamma \backslash G$.

We give a classification of birational transformations on smooth projective surfaces which have a Zariski-dense set of noncritical periodic points. In particular, we show that if the first dynamical degree is greater than one, the union of all noncritical periodic orbits is Zariski-dense.

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups and groups obtained by “random” ascending HNN (Higman–Neumann–Neumann) extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank $2$ free group sending $a$ to $ab$ and $b$ to $ba$; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension $2$) of a double of a free group along a collection of subgroups is a finite-sided rational polyhedron and that every rational class is virtually represented by an extremal surface subgroup. These results are obtained by a mixture of combinatorial, geometric, and linear programming techniques.