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Given a general pseudo-Anosov flow in a closed three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We construct a natural compactification of this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to a suspension Anosov flow, we then show: The ideal circle of the orbit space has a natural quotient space which is a sphere. This sphere is a dynamical systems ideal boundary for a compactification of the universal cover of the manifold. The main result is that the fundamental group acts on the flow ideal boundary as a uniform convergence group. Using a theorem of Bowditch, this yields a proof that the fundamental group of the manifold is Gromov hyperbolic and it shows that the action of the fundamental group on the flow ideal boundary is conjugate to the action on the Gromov ideal boundary. This gives an entirely new proof that the fundamental group of a closed, atoroidal –manifold which fibers over the circle is Gromov hyperbolic. In addition with further geometric analysis, the main result also implies that pseudo-Anosov flows without perfect fits are quasigeodesic flows and that the stable/unstable foliations of these flows are quasi-isometric foliations. Finally we apply these results to (nonsingular) foliations: if a foliation is –covered or with one sided branching in an aspherical, atoroidal three manifold then the results above imply that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity.
Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the genus mapping class group has a finite generating set whose size grows cubically with respect to . Our main tool is a new space called the handle graph on which acts cocompactly.
We fill a gap in the proof of the transversality result for quilted Floer trajectories in [Geom. Topol. 14 (2010) 833–902] by addressing trajectories for which some but not all components are constant. Namely we show that for generic sets of split Hamiltonian perturbations and split almost complex structures, the moduli spaces of parametrized quilted Floer trajectories of a given index are smooth of expected dimension. An additional benefit of the generic split Hamiltonian perturbations is that they perturb the given cyclic Lagrangian correspondence such that any geometric composition of its factors is transverse and hence immersed.
The aim of the article is an extension of the Monodromy Conjecture of Denef and Loeser in dimension two, incorporating zeta functions with differential forms and targeting all monodromy eigenvalues, and also considering singular ambient spaces. That is, we treat in a conceptual unity the poles of the (generalized) topological zeta function and the monodromy eigenvalues associated with an analytic germ defined on a normal surface singularity . The article targets the “right” extension in the case when the link of is a homology sphere. As a first step, we prove a splice decomposition formula for the topological zeta function for any and analytic differential form , which will play the key technical localization tool in the later definitions and proofs.
Then, we define a set of “allowed” differential forms via a local restriction along each splice component. For plane curves we show the following three guiding properties: (1) if is any pole of with allowed, then is a monodromy eigenvalue of , (2) the “standard” form is allowed, (3) every monodromy eigenvalue of is obtained as in (1) for some allowed and some .
For general we prove (1) unconditionally, and (2)–(3) under an additional (necessary) assumption, which generalizes the semigroup condition of Neumann–Wahl. Several examples illustrate the definitions and support the basic assumptions.
The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in a forthcoming paper.
In the Appendix written by S Ganatra and M Maydanskiy it is shown that the results of this paper imply P Seidel’s conjecture from [Proc. Sympos. Pure Math. 80, Amer. Math. Soc. (2009) 415–434].
We develop a new framework for cohomology of discrete metric spaces and groups which simultaneously generalises group cohomology, Roe’s coarse cohomology, Gersten’s –cohomology and Johnson’s bounded cohomology. In this framework we give an answer to Higson’s question concerning the existence of a cohomological characterisation of Yu’s property A, analogous to Johnson’s characterisation of amenability. In particular, we introduce an analogue of invariant mean for metric spaces with property A. As an application we extend Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This provides an alternative proof of Nowak’s result that the infinite dimensional cube does not have property A.
The decomposition theorem for smooth projective morphisms says that decomposes as . We describe simple examples where it is not possible to have such a decomposition compatible with cup product, even after restriction to Zariski dense open sets of . We prove however that this is always possible for families of surfaces (after shrinking the base), and show how this result relates to a result by Beauville and the author [J. Algebraic Geom. 13 (2004) 417–426] on the Chow ring of a surface . We give two proofs of this result, the first one involving –autocorrespondences of surfaces, seen as analogues of isogenies of abelian varieties, the second one involving a certain decomposition of the small diagonal in obtained by Beauville and the author. We also prove an analogue of such a decomposition of the small diagonal in for Calabi–Yau hypersurfaces in , which in turn provides strong restrictions on their Chow ring.
Let be the minimal resolution of the type surface singularity. We study the equivariant orbifold Gromov–Witten theory of the –fold symmetric product stack of . We calculate the divisor operators, which turn out to determine the entire theory under a nondegeneracy hypothesis. This, together with the results of Maulik and Oblomkov, shows that the Crepant Resolution Conjecture for is valid. More strikingly, we complete a tetrahedron of equivalences relating the Gromov–Witten theories of and the relative Gromov–Witten/Donaldson–Thomas theories of .
We prove that the asymptotic dimension of a finite-dimensional cube complex is bounded above by the dimension. To achieve this we prove a controlled colouring theorem for the complex. We also show that every cube complex is a contractive retraction of an infinite dimensional cube. As an example of the dimension theorem we obtain bounds on the asymptotic dimension of small cancellation groups.
In his study of the group of homology cylinders, J Levine [Algebr. Geom. Topol. 2 (2002) 1197–1204] made the conjecture that a certain group homomorphism is an isomorphism. Both and are defined combinatorially using trivalent trees and have strong connections to a variety of topological settings, including the mapping class group, homology cylinders, finite type invariants, Whitney tower intersection theory and the homology of . In this paper, we confirm Levine’s conjecture by applying discrete Morse theory to certain subcomplexes of a Kontsevich-type graph complex. These are chain complexes generated by trees, and we identify particular homology groups of them with the domain and range of Levine’s map.
The isomorphism is a key to classifying the structure of links up to grope and Whitney tower concordance, as explained in [Proc. Natl. Acad. Sci. USA 108 (2011) 8131–8138; arXiv 1202.3463]. In this paper and [arXiv 1202.2482] we apply our result to confirm and improve upon Levine’s conjectured relation between two filtrations of the group of homology cylinders.