We study the long time behaviour of the Ricci flow with bubbling-off on a possibly noncompact 3-manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston’s hyperbolisation theorem for 3-manifolds with toral boundary that generalises Perelman’s proof of the hyperbolisation conjecture in the closed case.

For a number field K with absolute Galois group G_K , we consider the action of G_K on the infinite tree of preimages of \alpha \in K under a degree-two rational function \phi \in K(x) , with particular attention to the case when \phi commutes with a non-trivial Möbius transformation. In a sense this is a dynamical systems analogue to the \ell -adic Galois representation attached to an elliptic curve, with particular attention to the CM case. Using a result about the discriminants of numerators of iterates of \phi , we give a criterion for the image of the action to be as large as possible. This criterion is in terms of the arithmetic of the forward orbits of the two critical points of \phi . In the case where \phi commutes with a non-trivial Möbius transformation, there is in effect only one critical orbit, and we give a modified version of our maximality criterion. We prove a Serre-type finite-index result in many cases of this latter setting.

This paper proves local-global principles for Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. We show in particular that such principles hold for H^n(F, \mathbb Z/m \mathbb Z(n-1)) , for all n>1 . This is motivated by work of Kato and others, where such principles were shown in related cases for n=3 . Using our results in combination with cohomological invariants, we obtain local-global principles for torsors and related algebraic structures over F . Our arguments rely on ideas from patching as well as the Bloch–Kato conjecture.

A foliation is \mathbb{R} -covered if the leaf space of the lifted foliation to the universal cover is homeomorphic to the set of real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to a fixed \mathbb{R} -covered foliation. If there are two transverse pseudo-Anosov flows, then the foliation is weakly conjugate to the stable foliation of an \mathbb{R} -covered Anosov flow. The proof uses the universal circle for \mathbb{R} -covered foliations.

We give examples of CAT(0), biautomatic, free-by-cyclic, one-relator groups which have finite-rank free subgroups of huge (Ackermannian) distortion. This leads to elementary examples of groups whose Dehn functions are similarly extravagant. This behaviour originates in manifestations of Hercules-versus-the-hydra battles in string-rewriting.

We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct, e.g., new groups having some interesting asphericity properties.

We study conformal deformations of a uniform space that satisfies the Ahlfors Q -regularity condition on balls of Whitney type. We verify the Gehring–Hayman Theorem by using a Whitney covering of the space.

In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simply-connected ones. We consider the isometric deformability question for an immersion x\colon M \to \mathbb R^3 of an oriented non-simply-connected surface with constant mean curvature H . We prove that the space of all isometric immersions of M with constant mean curvature H is, modulo congruences of \mathbb R^3 , either finite or a circle (Theorem 1.1). When it is a circle then, for the immersion x , every cycle in M has vanishing force and, when H\neq 0 , also vanishing torque. Moreover, we identify closed vector-valued 1-forms whose periods give the force and torque. Our work generalizes a classical result for minimal surfaces to constant mean curvature surfaces.

If K is a rationally null-homologous knot in a 3 -manifold M , the rational genus of K is the infimum of -\chi(S)/2p over all embedded orientable surfaces S in the complement of K whose boundary wraps p times around K for some p (hereafter: S is a p -Seifert surface for K ). Knots with very small rational genus can be constructed by “generic” Dehn filling, and are therefore extremely plentiful. In this paper we show that knots with rational genus less than 1/402 are all geometric – i.e. they may be isotoped into a special form with respect to the geometric decomposition of M – and give a complete classification. Our arguments are a mixture of hyperbolic geometry, combinatorics, and a careful study of the interaction of small p -Seifert surfaces with essential subsurfaces in M of non-negative Euler characteristic.

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are biLipschitz with respect to the visual metric.