If M is a hyperbolic 3-manifold fibering over the circle, the fundamental group of M acts faithfully by homeomorphisms on a circle (the circle at infinity of the universal cover of the fiber), preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures (e.g., taut foliations, quasigeodesic or pseudo-Anosov flows) are known to give rise to universal circles - a circle with a faithful pi(1)(M)-action preserving a pair of invariant laminations-and these universal circles play a key role in relating the dynamical structure to the geometry of M. In this paper we introduce the idea of zippers, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers (and their associated universal circles) may be constructed directly from uniform quasimorphisms or from actions.
We correct an error in the paper referred to in the title. Part of the argument is organized as a general method for establishing when (derived) functors factor through a fixed Serre subcategory, which may be of some more general interest.
We establish a connection between the theory of Ulrich sheaves and A1-homotopy theory. For instance, we prove that the A1-degree of a morphism between projective varieties, that is relatively oriented by an Ulrich sheaf, is constant on the target even when it is not A1-chain connected or A1-connected. Further if an embedded projective variety is the support of a symmetric Ulrich sheaf of rank one, the A1-degree of all its linear projections can be read off in an explicit way from the free resolution of the Ulrich sheaf. Finally, we construct an Ulrich sheaf on the secant variety of a curve and use this to define an arithmetic version of Viro's encomplexed writhe for curves in P3. This can be considered to be an arithmetic analogue of a knot invariant. Namely, we define a notion of algebraic isotopy under which the arithmetic writhe is invariant. For rational curves of degree at most four in P3 we obtain a complete classification up to algebraic isotopies.
For a group G of type FP(R) for a ring R and a homomorphism chi: G -> Z, we show that cd(R)(ker chi) = cd(R)(G)-1 if the top-dimensional cohomology of G with coefficients in the Novikov rings (R[G]) over cap (+/-chi) vanishes. This criterion is applied to show that if G is a finitely generated RFRS group of cohomological dimension 2, then G is virtually free-by-cyclic if and only if b(2)((2)) (G) = 0. This answers a question of Wise, and generalises and gives a significantly shorter proof of a recent theorem of Kielak and Linton, where the same result is obtained under the additional hypotheses that G is virtually compact special and hyperbolic. A consequence of the result is that all virtually RFRS groups of rational cohomological dimension 2 with vanishing second l(2)-Betti number are coherent. More generally, we show that if G is a RFRS group of cohomological dimension n and of type FPn-1(Q), then G admits a virtual map to Z with kernel of rational cohomological dimension n-1 if and only if b(n)((2)) (G) = 0.
Let X be a compact Gromov-Hausdorff limit space of a sequence of compact n-manifolds, M-i, of Ricci curvature Ric(Mi) >= - (n-1), volume vol(M-i) -> 0, and where all points in M-i are(delta, rho)-local rewinding Reifenberg points, or the sectional curvature satisfies sec M-i >=-1, respectively. We conjecture that if M-i is an aspherical manifold of fundamental group satisfying a certain condition (e.g., a nilpotent group), then X is a differentiable, or topological aspherical manifold, respectively. Our main result asserts that M-i is diffeomorphic or homeomorphic to a nilmanifold, then X is diffeomorphic or homeomorphic nilmanifold, respectively.
