In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most 1, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston’s conjecture. Here counter-examples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.

The main results of this paper involve general algebraic differentials ω on a general pencil of algebraic curves. We show how to determine whether ω is integrable in elementary terms for infinitely many members of the pencil. In particular, this corrects an assertion of James Davenport from 1981 and provides the first proof, even in rather strengthened form. We also indicate analogies with work of Andre and Hrushovski and with the Grothendieck–Katz Conjecture. To reach this goal, we first provide proofs of independent results which extend conclusions of relative Manin–Mumford type allied to the Zilber–Pink conjectures: we characterise torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2. In turn, we present yet another application of the latter results to a rather general pencil of Pell equations A^2−DB^2=1 over a polynomial ring. We determine whether the Pell equation (with squarefree D) is solvable for infinitely many members of the pencil.

We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.

We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with exotic properties: • We construct the first examples of finitely generated coarsely non-amenable groups (that is, groups without Guoliang Yu’s Property A) that are coarsely embeddable into a Hilbert space. Moreover, our groups act properly on CAT(0) cubical complexes. • We construct the first examples of finitely generated groups, with expanders embedded isometrically into their Cayley graphs—in contrast, in the case of the Gromov monster expanders are not even coarsely embedded. We present further applications.

An irreducible integrable connection (E,∇) on a smooth projective complex variety X is called rigid if it gives rise to an isolated point of the corresponding moduli space M_{dR}(X). According to Simpson’s motivicity conjecture, irreducible rigid flat connections are of geometric origin, that is, arise as subquotients of a Gauss-Manin connection of a family of smooth projective varieties defined on an open dense subvariety of X. In this article we study mod-p reductions of irreducible rigid connections and establish results which confirm Simpson’s prediction. In particular, for large p, we prove that p-curvatures of mod-p reductions of irreducible rigid flat connections are nilpotent, and building on this result, we construct an F-isocrystalline realization for irreducible rigid flat connections. More precisely, we prove that there exist smooth models X_R and (E_R,∇_R) of X and (E,∇), over a finite-type ring R, such that for every Witt ring W(k) of a finite field k and every homomorphism R→W(k), the p-adic completion of the base change (\hat E_{W(k)}, \hat ∇_{W(k)}) on \hat X_{W(k)} represents an F-isocrystal. Subsequently, we show that irreducible rigid flat connections with vanishing p-curvatures are unitary. This allows us to prove new cases of the Grothendieck–Katz p-curvature conjecture. We also prove the existence of a complete companion correspondence for F-isocrystals stemming from irreducible cohomologically rigid connections.