Universal circles for quasigeodesic flows

We consider the problem of when a closed orientable hyperbolic surface admits a totally geodesic embedding into a closed orientable hyperbolic 3-manifold; given a finite isometric group action on the surface, we consider also an equivariant version of such an embedding. We prove that an equivariant embedding exists for all finite irreducible group actions on surfaces; such surfaces are known also as quasiplatonic surfaces; in particuler, all quasiplatonic surfaces embed geode...

In this article, we show that, for any compact 3-manifold, there is a $C^{1}$ volume-minimizing one-dimensional foliation. More generally, we show the existence of mass-minimizing rectifiable sections of sphere bundles without isolated "pole points" in the base manifold. This same analysis is used to show that the examples, due to Sharon Pedersen, of potentially volume-minimizing rectifiable sections (rectifiable foliations) of the unit tangent bundle to $S^{2n+1}$ are not, i...

In 1978, W. Thurston revolutionized low diemsional topology with his work on hyperbolic 3-manifolds. In this paper, we discuss what is currently known about knots in the 3-sphere with hyperbolic complements. Then focus is on geometric invariants coming out of the hyperbolic structures. This is one of a collection of articles to appear in the Handbook of Knot Theory.

We show that Thurston's earthquake flow is strongly asymmetric in the sense that its normalizer is as small as possible inside the group of orbifold automorphisms of the bundle of measured geodesic laminations over moduli space. (At the level of Teichm\"uller space, such automorphisms correspond to homeomorphisms that are equivariant with respect to an automorphism of the mapping class group.) It follows that the earthquake flow does not extend to an $\mathrm{SL}(2,\mathbf{R}...

We consider the problem of when a closed hyperbolic surface admits a totally geodesic embedding into a closed hyperbolic 3-manifold, and in particular equivariant versions of such embeddings. In a previous paper we considered orientation-preserving actions on orientable surfaces; in the present paper, we consider large orientation-reversing actions on orientable surfaces, and also large actions on nonorientable surfaces.

For closed hyperbolic $3$-manifolds $M$, Brock and Dunfield prove an inequality on the first cohomology bounding the ratio of the geometric $L^2$-norm to the topological Thurston norm. Motivated by Dehn fillings, they conjecture that as the injectivity radius tends to $0$, the ratio is big O of the square root of the log of the injectivity radius. We prove this conjecture for all sequences of manifolds which geometrically converge. Generically, we prove that the ratio is boun...

We analyse the existence question for essential laminations in 3-manifolds. The purpose is to prove that there are infinitely many closed hyperbolic 3-manifolds which do not admit essential laminations. This answers in the negative a question posed by Gabai and Oertel. The proof is obtained by analysing certain group actions on trees and showing that certain 3-manifold groups only have trivial actions on trees. In general the trees are neither simplicial nor metric. There are...

We study the dynamics of the geodesic and horocycle flows of the unit tangent bundle $(\hat M, T^1\mathcal{F})$ of a compact minimal lamination $(M,\mathcal F)$ by negatively curved surfaces. We give conditions under which the action of the affine group generated by the joint action of these flows is minimal, and examples where this action is not minimal. In the first case, we prove that if $\mathcal F$ has a leaf which is not simply connected, the horocyle flow is topologica...

We apply an equivariant version of Perelman's Ricci flow with surgery to study smooth actions by finite groups on closed 3-manifolds. Our main result is that such actions on elliptic and hyperbolic 3-manifolds are conjugate to isometric actions. Combining our results with results by Meeks and Scott [17], it follows that such actions on geometric 3-manifolds (in the sense of Thurston) are always geometric, i.e. there exist invariant locally homogeneous Riemannian metrics. This...

We study R-covered foliations of 3-manifolds from the point of view of their transverse geometry. For an R-covered foliation in an atoroidal 3-manifold M, we show that M-tilde can be partially compactified by a canonical cylinder S^1_univ x R on which pi_1(M) acts by elements of Homeo(S^1) x Homeo(R), where the S^1 factor is canonically identified with the circle at infinity of each leaf of F-tilde. We construct a pair of very full genuine laminations transverse to each other...