poincare bendixson theory for leaves of codimension one the level of a local minimal set of a c2 codimension one foliation of a compact manifold is a nonnegative integer defined inductively level zero corresponding to the minimal sets in the usual sense each leaf of a local minimal set at level k is at level k the authors develop a theory of local minimal sets level and how leaves at level k asymptotically approach leaves at lower level this last generalizes the classical poincare bendixson theorem and provides information relating growth topological type and level e g if l is a totally proper leaf at level k then l has exactly polynomial growth of degree k and topological type k

cusp geometry of fibered 3 manifolds let f be a surface and suppose that varphi colon f to f is a pseudo anosov homeomorphism fixing a puncture p of f the mapping torus m m varphi is hyperbolic and contains a maximal cusp c about the puncture p we show that the area and height of the cusp torus partial c is equal to the stable translation distance of varphi acting on the arc complex scr a f p up to an explicitly bounded multiplicative error our proof relies on elementary facts about the hyperbolic geometry of pleated surfaces in particular the proof of this theorem does not use any deep results from teichm uller theory kleinian group theory or the coarse geometry of scr a f p a similar result holds for quasi fuchsian manifolds n cong f times bbb r in that setting we find a combinatorial estimate for the area and height of the cusp annulus in the convex core of n up to explicitly bounded multiplicative and additive error as an application we show that covers of punctured surfaces induce quasi isometric embeddings of arc complexes

non lerfness of arithmetic hyperbolic manifold groups and mixed 3 manifold groups we will show that for any noncompact arithmetic hyperbolic m manifold with m 3 and any compact arithmetic hyperbolic m manifold with m 4 that is not a 7 dimensional one defined by octonions its fundamental group is not locally extended residually finite lerf the main ingredient in the proof is a study on abelian amalgamations of hyperbolic 3 manifold groups we will also show that a compact orientable irreducible 3 manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is lerf

separable subsets of gferf negatively curved groups a word hyperbolic group g is called gferf if every quasiconvex subgroup coincides with the intersection of finite index subgroups containing it we show that in any such group the product of finitely many quasiconvex subgroups is closed in the profinite topology on g

free abelian covers short loops stable length and systolic inequalities we explore the geometry of the abel jacobi map f from a closed orientable riemannian manifold x to its jacobi torus h 1 x mathbb r h 1 x mathbb z mathbb r applying m gromov s filling inequality to the typical fiber of f we prove an interpolating inequality for two flavors of shortest length invariants of loops the procedure works provided the lift of the fiber is non trivial in the homology of the maximal free abelian cover overline x classified by f we show that the finite dimensionality of the rational homology of overline x is a sufficient condition for the homological non triviality of the fiber when applied to nilmanifolds our fiberwise inequality typically gives stronger information than the filling inequality for x itself in dimension 3 we present a sufficient non vanishing condition in terms of massey products this condition holds for certain manifolds that do not fiber over their jacobi torus such as 0 framed surgeries on suitable links our systolic inequality applies to surface bundles over the circle provided the algebraic monodromy has 1 dimensional coinvariants even though the massey product invariant vanishes for some of these bundles

essential closed surfaces in bounded 3 manifolds a question dating back to waldhausen 10 and discussed in various contexts by thurston see 9 is the problem of the extent to which irreducible 3 manifolds with infinite fundamental group must contain surface groups to state our results precisely it is convenient to make the definition that a map i s 9 m of a closed orientable connected surface s is essential if it is injective at the level of fundamental groups and the group i rr1 s cannot be conjugated into a subgroup 7rr com of rr m where om is a component of om this latter condition is equivalent to the statement that the image of the surface s cannot be freely homotoped into om one of the main results of this paper is the following

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

foliations with good geometry the goal of this article is to show that there is a large class of closed hyperbolic 3 manifolds admitting codimension one foliations with good large scale geometric properties we obtain results in two directions first there is an internal result a possibly singular foliation in a manifold is quasi isometric if when lifted to the universal cover distance along leaves is efficienit up to a bounded multiplicative distortion in measuring distance in the universal cover this means that leaves reflect very well the geometry in the large of the universal cover and are geometrically tight this is the best geometric behavior we previously proved that nonsingular codimension one foliations in closed hyperbolic 3 manifolds can never be quasi isometric in this article we produce a large class of singular quasi isometric codimension one foliations in closed hyperbolic 3 manifolds the foliations are stable and unstable foliations of pseudo anosov flows our second result is an external result relating nonsingular foliations in hyperbolic 3 manifolds with their limit sets in the universal cover that is showing that leaves in the universal cover have good asymptotic behavior let g be a reebless finite depth foliation in a closed hyperbolic 3 manifold then g is not quasi isometric but suppose that g is transverse to a quasigeodesic pseudo anosov flow with quasi isometric stable and unstable foliations which are given by the internal result we then show that the lifts of leaves of g to the universal cover extend continuously to the sphere at infinity and we also produce infinitely many examples satisfying the hypothesis the main tools used to prove these results are first a link between geometric properties of stable unstable foliations of pseudo anosov flows and the topology of these foliations in the tuniversal cover and second a topological theory of the joint structure of the pseudo anosov foliations in the universal cover reebless codimension one foliations are extremely useful for understanding the topology of 3 manifolds for instance they imply that the manifold is irreducible ro its universal cover is homeomorphic to r3 pa leaves are 7rl injective no and transversals are never null homotopic no hence they reflect topological properties of the manifold as for which manifolds have reebless foliations gabai gal ga2 ga3 proved that any compact oriented irreducible 3 manifold with nonzero first betti number has many reebless finite depth foliations roughly a

a general regularity theory for stable codimension 1 integral varifolds we give a necessary and sufficient geometric structural condition which we call the a structural hypothesis for a stable codimension 1 integral varifold on a smooth riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities the a structural hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded c 1 a hypersurfaces with boundary meeting only along their common boundary we establish that whenever a stable integral n varifold on a smooth n 1 dimensional riemannian manifold satisfies the a structural hypothesis for some a 0 1 2 its singular set is empty if n 6 discrete if n 7 and has hausdorff dimension n 7 if n 8 in view of well known examples this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses we also establish compactness of mass bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a structural hypothesis for some a 0 1 2 the a structural hypothesis on an n varifold for any a 0 1 2 is readily implied by either of the following two hypotheses i the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary ii the singular set of the varifold has vanishing n 1 dimensional hausdorff measure thus our theory subsumes the well known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions an optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

minimal genus in 4 manifolds with a free circle action abstract let n be a closed irreducible 3 manifold and assume n is not a graph manifold we improve for all but finitely many s 1 bundles m over n the adjunction inequality for the minimal complexity of embedded surfaces this allows us to completely determine the minimal complexity of embedded surfaces in all but finitely many s 1 bundles over a large class of 3 manifolds

l2 topological invariants of 3 manifolds we give results on thel2 betti numbers and novikov shubin invariants of compact manifolds especially 3 manifolds we first study the betti numbers and novikov shubin invariants of a chain complex of hilbert modules over a finite von neumann algebra we establish inequalities among the novikov shubin invariants of the terms in a short exact sequence of chain complexes our algebraic results along with some analytic results on geometric 3 manifolds are used to compute thel2 betti numbers of compact 3 manifolds which satisfy a weak form of the geometrization conjecture and to compute or estimate their novikov shubin invariants

on numerical boundary treatment of hyperbolic systems for finite difference and finite element methods it is well known that the stability of the initial boundary value problem for a scalar equation does not necessarily imply stability for a vector equation with a similar boundary treatment in fact we show that for any boundary treatment one can construct systems for which the boundary treatment on the natural variables is not stable our main theorem is that one can introduce a treatment of the boundaries for which the stability of the system follows immediately from the stability for a scalar equation this is accomplished by operating on the characteristic variables for those quantities that are not specified on the boundary in a working code this can be accomplished by adding a correction term to the existing boundary algorithm the analysis and computational results are presented for both finite differences and semi discrete galerkin methods

a two dimensional time dependent chebyshev method of collocation for the study of astrophysical flows abstract we present a two dimensional time dependent collocation chebyshev spectral method as a tool for the solution of certain astrophysical flows in this paper we solve the navier stokes equations in spherical coordinates the problem of the propagation of an initial perturbation in an atmosphere of a compact star is used to demonstrate the method and its power in particular viscosity and rotation are included

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

mixed 3 manifolds are virtually special letm be a compact oriented irreducible 3 manifold which is neither a graph manifold nor a hyperbolic manifold we prove that 1m is virtually special

degree of l 2 alexander torsion for 3 manifolds for an irreducible orientable compact 3 manifold n with empty or incompressible toral boundary the full l 2 alexander torsion tau 2 n phi t associated to any real first cohomology class phi of n is represented by a function of a positive real variable t the paper shows that tau 2 n phi is continuous everywhere positive and asymptotically monomial in both ends moreover the degree of tau 2 n phi equals the thurston norm of phi the result confirms a conjecture of j dubois s friedl and w luck and addresses a question of w li and w zhang associated to any admissible homomorphism gamma pi 1 n rightarrow g the l 2 alexander torsion tau 2 n gamma phi is shown to be continuous and everywhere positive provided that g is residually finite and n gamma is weakly acyclic in this case a generalized degree can be assigned to tau 2 n gamma phi moreover the generalized degree is bounded by the thurston norm of phi

on locally indicable groups a group is said to be locally indicable if each of its non trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image such groups were studied by higman 9 in connection with the zero divisor and unit problems for group rings more recently they have arisen i 2 12 in the study of equations over groups in 12 i proved a freiheitsatz for locally indicable groups this has been proved independently by brodskii 2 and short 22 the present paper arises from an investigation of a question put to me by s j pride whether torsionfree 1 relator groups are locally indicable the question was raised originally by baumslag i 1 problem 19 and an affirmative solution has recently been announced by brodski 2 as a consequence the group algebra rg of a torsion free 1 relator group g over an integral domain r has no non trivial zero divisors and no non trivial units using higman s results 9 the first fact was also proved by lewin and lewin 16 by embedding rg in a division ring the second appears to be new as was pointed out to me by k a brown a second consequence is that no 1 relator group has a non trivial finitely generated perfect subgroup which answers 1 problem 7 and 10 question 1 in fact using the freiheitsatz and the tower method described in 12 it is possible to prove the following general version of brodskii s theorem

coxeter groups are virtually special abstract in this paper we prove that every finitely generated coxeter group has a finite index subgroup that is the fundamental group of a special cube complex some consequences include every f g coxeter group is virtually a subgroup of a right angled coxeter group every word hyperbolic coxeter group has separable quasiconvex subgroups

unknotting tunnels and seifert surfaces let k be a knot with an unknotting tunnel gamma and suppose that k is not a 2 bridge knot there is an invariant rho p q in mathbb q 2 mathbb z with p odd defined for the pair k gamma the invariant rho has interesting geometric properties it is often straightforward to calculate for example for k a torus knot and gamma an annulus spanning arc rho k gamma 1 although rho is defined abstractly it is naturally revealed when k cup gamma is put in thin position if rho neq 1 then there is a minimal genus seifert surface f for k such that the tunnel gamma can be slid and isotoped to lie on f one consequence is that if rho k gamma neq 1 then mathrm genus k 1 this confirms a conjecture of goda and teragaito for pairs k gamma with rho k gamma neq 1

knot floer homology detects genus one fibred knots ozsv ath and szab o conjectured that knot floer homology detects fibred knots we propose a strategy to approach this conjecture based on contact topology and gabai s theory of sutured manifold decomposition we implement this strategy for genus one knots obtaining as a corollary that if rational surgery on a knot k gives the poincar e homology sphere sigma 2 3 5 then k is the left handed trefoil knot

optimal time splitting for two and three dimensional navier stokes equations with mixed derivatives a new explicit time splitting algorithm has been developed for finite difference modelling of the full two and three dimensional time dependent compressible viscous navier stokes equations of fluid mechanics the scheme is optimal in the sense that the split operators achieve their maximum allowable time step i e the corresponding courant number the algorithm allows a conservation form formulation stability is proven analytically and verified numerically in proving stability it was shown that all nine matrix coefficients of the navier stokes equations are simultaneously symmetrizable by a similarity transformation two such transformations and their resulting symmetric matrix coefficients are presented explicitly