Abstract: The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have eventually quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is eventually a ratio of quasipolynomials, and that unit balls in the scl norm eventually quasiconverge in finite-dimensional surgery families.

Publication: Transactions of the American Mathematical Society Vol.: 365ISSN: 0002-9947

ID: CaltechAUTHORS:20131024-153105928

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Abstract: If K is a rationally null-homologous knot in a 3-manifold M, the rational genus of K is the infimum of _(−χ)(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.

Publication: Commetarii Mathematici Helvetici Vol.: 88 No.: 1 ISSN: 0010-2571

ID: CaltechAUTHORS:20130528-092038076

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Abstract: These notes are a self-contained introduction to the use of dynamical and probabilistic methods in the study of hyperbolic groups. Moat of this material is standard; however some of the proofs given are new, and some results are proved in greater generality than have appeared in the literature.

No.: 597
ID: CaltechAUTHORS:20131007-132219534

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Abstract: An arbitrary homomorphism between groups is nonincreasing for stable commutator length, and there are infinitely many (injective) homomorphisms between free groups which strictly decrease the stable commutator length of some elements. However, we show in this paper that a random homomorphism between free groups is almost surely an isometry for stable commutator length for every element; in particular, the unit ball in the scl norm of a free group admits an enormous number of exotic isometries. Using similar methods, we show that a random fatgraph in a free group is extremal (i.e., is an absolute minimizer for relative Gromov norm) for its boundary; this implies, for instance, that a random element of a free group with commutator length at most n has commutator length exactly n and stable commutator length exactly n-1/2. Our methods also let us construct explicit (and computable) quasimorphisms which certify these facts.

Publication: New York Journal of Mathematics Vol.: 17ISSN: 1076-9803

ID: CaltechAUTHORS:20120320-102304082

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Abstract: We establish the existence of new rigidity and rationality phenomena in the theory of nonabelian group actions on the circle and introduce tools to translate questions about the existence of actions with prescribed dynamics into finite combinatorics. A special case of our theory gives a very short new proof of Naimi's theorem (i.e., the conjecture of Jankins-Neumann) which was the last step in the classification of taut foliations of Seifert fibered spaces.

Publication: Journal of Modern Dynamics Vol.: 5 No.: 4 ISSN: 1930-5311

ID: CaltechAUTHORS:20120501-104756665

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Abstract: A hyperbolic conjugacy class in the modular group PSL(2, Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface.

Publication: Proceedings of the American Mathematical Society Vol.: 139 No.: 7 ISSN: 0002-9939

ID: CaltechAUTHORS:20110805-111542579

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Abstract: This paper initiates a systematic study of the relation of commensurability of surface automorphisms, or equivalently, fibered commensurability of 3-manifolds fibering over S^1. We show that every hyperbolic fibered commensurability class contains a unique minimal element. The situation for toroidal manifolds is more complicated, and we illustrate a range of phenomena that can occur in this context.

Publication: Pacific Journal of Mathematics Vol.: 250 No.: 2 ISSN: 0030-8730

ID: CaltechAUTHORS:20110418-113350448

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Abstract: We establish a close connection between the stable commutator length in free groups and the geometry of sails (roughly, the boundary of the convex hull of the set of integer lattice points) in integral polyhedral cones. This connection allows us to show that the scl norm is piecewise rational linear in free products of Abelian groups, and that it can be computed via integer programing. Furthermore, we show that the scl spectrum of non-Abelian free groups contains elements congruent to every rational number modulo ℤ, and contains well-ordered sequences of values with ordinal type ω^ω. Finally, we study families of elements w(p) in free groups obtained by surgery on a fixed element w in a free product of Abelian groups of higher rank, and show that scl(w(p)) → scl(w) as p → ∞.

Publication: Journal of Topology Vol.: 4 No.: 2 ISSN: 1753-8416

ID: CaltechAUTHORS:20110726-095541078

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Abstract: We study stable W-length in groups, especially for W equal to the n-fold commutator γ_n := [x_1, [x_2,...[x_(n−1), x_n]]....We prove that in any perfect group, for any n ≥ 2 and any element g, the stable commutator length of g is at least as big as 2^(2−n) times the stable γ_n-length of g. We also establish analogues of Bavard duality for words γn and for β_2 := [[x, y], [z,w]]. Our proofs make use of geometric properties of the asymptotic cones of verbal subgroups with respect to bi-invariant metrics. In particular, we show that for suitable W, these asymptotic cones contain certain subgroups that are normed vector spaces.

Publication: Contemporary Mathematics No.: 560 ISSN: 0271-4132

ID: CaltechAUTHORS:20120319-082257258

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Abstract: A function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include: (1) homomorphisms to Z; (2) word length with respect to a finite generating set; (3) most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms). We show that bicombable functions on word-hyperbolic groups satisfy a central limit theorem: if φ(overbar)_n is the value of φ on a random element of word length n (in a certain sense), there are E and σ for which there is convergence in the sense of distribution n^(-1/2)(φ(overbar)_n - nE)→ N(0,σ), where N(0,σ) denotes the normal distribution with standard deviation σ. As a corollary, we show that if S_1 and S_2 are any two finite generating sets for G, there is an algebraic number λ_(1,2) depending on S_1 and S_2 such that almost every word of length n in the S1 metric has word length n∙λ_(1,2) in the S_2 metric, with error of size O (√n).

Publication: Ergodic Theory and Dynamical Systems Vol.: 30 No.: 5 ISSN: 0143-3857

ID: CaltechAUTHORS:20101108-100431663

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Abstract: We give a short derivation of an identity of Martin Bridgeman concerning orthospectra of hyperbolic surfaces.

Publication: Topology Proceedings Vol.: 38ISSN: 0146-4124

ID: CaltechAUTHORS:20111109-113827923

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Abstract: We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd integer n, there is a rational function q_n of degree 2(n − 2) so that if M is a compact hyperbolic manifold of dimension n with totally geodesic boundary S, there is an identity χ(S) = ∑_(i) q_(n)(e^l_i) where the sum is taken over the orthospectrum of M. When n = 3, this has the explicit form ∑_(i) 1/(e^(2l_(i)) − 1) = −χ(S)/4.

Publication: Algebraic and Geometric Topology Vol.: 10 No.: 3 ISSN: 1472-2747

ID: CaltechAUTHORS:20101122-112850841

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Abstract: Stable commutator length vanishes in any group that obeys a law.

Publication: Algebraic and Geometric Topology Vol.: 10 No.: 1 ISSN: 1472-2747

ID: CaltechAUTHORS:20100524-132519728

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Abstract: Associated to a closed, oriented surface S is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary (2+1)-dimensional TQFTs. The proof involves the construction of a suitable complexity function c on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c(AB) ≤max(c(AA),c(BB)) for all A,B which bound S, with equality if and only if A=B. The complexity function c involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol-Storm-Thurston.

Publication: Journal of the American Mathematical Society Vol.: 23 No.: 1 ISSN: 0894-0347

ID: CaltechAUTHORS:20100201-095353095

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Abstract: In this paper we obtain uniform positive lower bounds on the stable commutator length of elements in word-hyperbolic groups and certain groups acting on hyperbolic spaces (namely the mapping class group acting on the complex of curves, and an amalgamated free product acting on an associated Bass-Serre tree). If G is a word-hyperbolic group that is δ-hyperbolic with respect to a symmetric generating set S, then there is a positive constant C depending only on δ and on |S| such that every element of G either has a power which is conjugate to its inverse, or else the stable commutator length of the element is at least equal to C. By Bavard’s theorem, these lower bounds on stable commutator length imply the existence of quasimorphisms with uniform control on the defects; however, we show how to construct such quasimorphisms directly. We also prove various separation theorems on families of elements in such groups, constructing homogeneous quasimorphisms (again with uniform estimates) which are positive on some prescribed element while vanishing on some family of independent elements whose translation lengths are uniformly bounded. Finally, we prove that the first accumulation point for stable commutator length in a torsion-free word-hyperbolic group is contained between 1/12 and 1/2. This gives a universal sense of what it means for a conjugacy class in a hyperbolic group to have a small stable commutator length, and can be thought of as a kind of “homological Margulis lemma”.

Publication: Groups, Geometry, and Dynamics Vol.: 4 No.: 1 ISSN: 1661-7207

ID: CaltechAUTHORS:20100107-135614705

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Abstract: For any group, there is a natural (pseudo-)norm on the vector space B^H_1 of real homogenized (group) 1 -boundaries, called the stable commutator length norm. This norm is closely related to, and can be thought of as a relative version of, the Gromov (pseudo)-norm on (ordinary) homology. We show that for a free group, the unit ball of this pseudo-norm is a rational polyhedron. It follows that the stable commutator length in free groups takes on only rational values. Moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group. The proof of these facts yields an algorithm to compute the stable commutator length in free groups. Using this algorithm, we answer a well-known question of Bavard in the negative, constructing explicit examples of elements in free groups whose stable commutator length is not a half-integer.

Publication: Journal of the American Mathematical Society Vol.: 22 No.: 4 ISSN: 0894-0347

ID: CaltechAUTHORS:20091020-133808896

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Abstract: Let F = π_1(S) where S is a compact, connected, oriented surface with χ(S) < 0 and nonempty boundary. (1) The projective class of the chain ∂S ∈ B^H_1(F) intersects the interior of a codimension one face π_S of the unit ball in the stable commutator length norm on B^H_1(F). (2) The unique homogeneous quasimorphism on F dual to π_S (up to scale and elements of H^1(F)) is the rotation quasimorphism associated to the action of π_1(S) on the ideal boundary of the hyperbolic plane, coming from a hyperbolic structure on S. These facts follow from the fact that every homologically trivial 1–chain C in S rationally cobounds an immersed surface with a sufficiently large multiple of ∂S. This is true even if S has no boundary.

Publication: Geometry and Topology Vol.: 13 No.: 3 ISSN: 1465-3060

ID: CaltechAUTHORS:20090828-095911315

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Abstract: This book is a comprehensive introduction to the theory of stable commutator length, an important subfield of quantitative topology, with substantial connections to 2-manifolds, dynamics, geometric group theory, bounded cohomology, symplectic topology, and many other subjects. We use constructive methods whenever possible, and focus on fundamental and explicit examples. We give a self-contained presentation of several foundational results in the theory, including Bavard’s Duality Theorem, the Spectral Gap Theorem, the Rationality Theorem, and the Central Limit Theorem. The contents should be accessible to any mathematician interested in these subjects, and are presented with a minimal number of prerequisites, but with a view to applications in many areas of mathematics.

No.: 20
ID: CaltechAUTHORS:20180808-090434034

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Abstract: This is an exposition of the homological classification of actions of surface groups on the plane, in every degree of smoothness.

Vol.: 498
ID: CaltechAUTHORS:20100511-111051682

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Abstract: Let G be a finitely presented group, and G′ its commutator subgroup. Let C be the Cayley graph of G′ with all commutators in G as generators. Then C is large scale simply connected. Furthermore, if G is a torsion-free nonelementary word-hyperbolic group, C is one-ended. Hence (in this case), the asymptotic dimension of C is at least 2.

Publication: Algebraic & Geometric Topology Vol.: 8 No.: 4 ISSN: 1472-2747

ID: CaltechAUTHORS:20170408-171448676

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Abstract: Kronecker allegedly once said, “God created the natural numbers; all the rest is the work of man.” But to a topologist, the natural numbers are just a tool for classifying orientable surfaces, bycounting the number of handles (or genus).

Publication: Notices of the American Mathematical Society Vol.: 55 No.: 9 ISSN: 0002-9920

ID: CaltechAUTHORS:20180814-080518022

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Abstract: Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H2(G; ℚ) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov–Thurston norm on H2(G; ℝ) is a finite-sided rational polyhedron.

Publication: Geometry and Topology Vol.: 12 No.: 4 ISSN: 1465-3060

ID: CaltechAUTHORS:CALgt08

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Abstract: A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length of any element with a nonzero crossing number is positive, and bounded below by a constant depending only (and explicitly) on a bound on the crossing numbers of generating elements. This answers a question of Benson Farb.

Publication: Proceedings of the American Mathematical Society Vol.: 136 No.: 7 ISSN: 0002-9939

ID: CaltechAUTHORS:CALpams08

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Abstract: By the Thurston Stability Theorem, a group of C^1 orientation-preserving diffeomorphisms of the closed unit interval is locally indicable. We show that the local order structure of orbits gives a stronger criterion for nonsmoothability that can be used to produce new examples of locally indicable groups of homeomorphisms of the interval that are not conjugate to groups of C^1 diffeomorphisms.

Publication: Algebraic and Geometric Topology Vol.: 8 No.: 1 ISSN: 1472-2747

ID: CaltechAUTHORS:20090828-094355897

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Abstract: Let G be a subgroup of PL^+(I). Then the stable commutator length of every element of [G, G] is zero.

Publication: Pacific Journal of Mathematics Vol.: 232 No.: 2 ISSN: 0030-8730

ID: CaltechAUTHORS:20100112-081852801

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Abstract: We show that if M is a hyperbolic 3–manifold which admits a quasigeodesic flow, then π1(M) acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generalizing a theorem of Mosher, and we give the first example of a closed hyperbolic 3–manifold without a quasigeodesic flow, answering a long-standing question of Thurston.

Publication: Geometry and Topology Vol.: 10ISSN: 1465-3060

ID: CaltechAUTHORS:CALgt06c

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Abstract: If M is a hyperbolic once-punctured torus bundle over S 1, then the trace field of M has no real places.

Publication: Geometriae Dedicata Vol.: 118 No.: 1 ISSN: 0046-5755

ID: CaltechAUTHORS:20201207-153418820

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Abstract: We give an example of a subgroup of SL_2 ℂ which is a strictly ascending HNN extension of a non-abelian finitely generated free group F. In particular, we exhibit a free group F in SL_2 ℂ of rank 6 which is conjugate to a proper subgroup of itself. This answers positively a question of Drutu and Sapir (2005). The main ingredient in our construction is a specific finite volume (non-compact) hyperbolic 3-manifold M which is a surface bundle over the circle. In particular, most of F comes from the fundamental group of a surface fiber. A key feature of M is that there is an element of π1(M) in SL_2 ℂ with an eigenvalue which is the square root of a rational integer. We also use the Bass-Serre tree of a field with a discrete valuation to show that the group F we construct is actually free.

Publication: Proceedings of the American Mathematical Society Vol.: 134 No.: 11 ISSN: 0002-9939

ID: CaltechAUTHORS:20100318-151445653

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Abstract: We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(Sn), thought of as a discrete group. An appendix by Y de Cornulier shows that Homeo(Sn) has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(Sn) on a metric space by isometries has bounded orbits.

Publication: Geometry and Topology Vol.: 10 No.: 7 ISSN: 1465-3060

ID: CaltechAUTHORS:CALgt06a

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Abstract: In this paper we introduce and study the notion of dynamical forcing. Basically, we develop a toolkit of techniques to produce finitely presented groups which can only act on the circle with certain prescribed dynamical properties. As an application, we show that the set X ⊂ R/Z consisting of rotation numbers θ which can be forced by finitely presented groups is an infinitely generated Q-module, containing countably infinitely many algebraically independent transcendental numbers. Here a rotation number θ is forced by a pair (G_θ, α), where G_θ is a finitely presented group G_θ and α ∈ G_θ is some element, if the set of rotation numbers of ρ(α) as ρ ∈ Hom(G_θ, Homeo^(+)(S^1)) is precisely the set {0,±θ}. We show that the set of subsets of R/Z which are of the form rot(X(G, α)) = {r ∈ R/Z | r = rot(ρ(α)), ρ ∈ Hom(G, Homeo^(+)(S^1))}, where G varies over countable groups, are exactly the set of closed subsets which contain 0 and are invariant under x→−x. Moreover, we show that every such subset can be approximated from above by rot(X(G_i, α_i)) for finitely presented G_i. As another application, we construct a finitely generated group Γ which acts faithfully on the circle, but which does not admit any faithful C^1 action, thus answering in the negative a question of John Franks.

Publication: Transactions of the American Mathematical Society Vol.: 358 No.: 8 ISSN: 0002-9947

ID: CaltechAUTHORS:20110120-094122587

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Abstract: 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.

Publication: Journal of the American Mathematical Society Vol.: 19 No.: 2 ISSN: 0894-0347

ID: CaltechAUTHORS:20090414-123644599

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Abstract: We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that certain generalized braid groups are circularly-orderable. We also show that the Euler class of C^infty diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus >1 admits a C^infty action with arbitrary Euler class. On the other hand, we show that Z oplus Z actions satisfy a homological rigidity property: every orientation-preserving C^1 action of Z oplus Z on the plane has trivial Euler class. This gives the complete homological classification of surface group actions on R^2 in every degree of smoothness.

No.: 7
ID: CaltechAUTHORS:CALgtm04

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Abstract: We show that every orientable 3-manifold is a classifying space B\Gamma where \Gamma is a groupoid of germs of homeomorphisms of R. This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. The F we construct can be taken to be C^1 but not C^2. The existence of such an F answers positively a question posed by Tsuboi [Classifying spaces for groupoid structures, notes from minicourse at PUC, Rio de Janeiro (2001)], but leaves open the question of whether M = B\Gamma for some C^\infty groupoid \Gamma.

Publication: Algebraic and Geometric Topology Vol.: 2 No.: 21 ISSN: 1472-2747

ID: CaltechAUTHORS:CALagt02

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Abstract: We show that every topological surface lamination of a 3-manifold M is isotopic to one with smoothly immersed leaves. This carries out a project proposed by Gabai in [Problems in foliations and laminations, AMS/IP Stud. Adv. Math. 2.2 1--33]. Consequently any such lamination admits the structure of a Riemann surface lamination, and therefore useful structure theorems of Candel [Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. 26 (1993) 489--516] and Ghys [Dynamique et geometrie complexes, Panoramas et Syntheses 8 (1999)] apply.

Publication: Algebraic and Geometric Topology Vol.: 1 No.: 29 ISSN: 1472-2747

ID: CaltechAUTHORS:CALagt01

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Abstract: 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 and to F, which bind every leaf of F. This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov flow for F, analogous to Thurston's structure theorem for surface bundles over a circle with pseudo-Anosov monodromy. A corollary of the existence of this structure is that the underlying manifold M is homotopy rigid in the sense that a self-homeomorphism homotopic to the identity is isotopic to the identity. Furthermore, the product structures at infinity are rigid under deformations of the foliation F through R-covered foliations, in the sense that the representations of pi_1(M) in Homeo((S^1_univ)_t) are all conjugate for a family parameterized by t. Another corollary is that the ambient manifold has word-hyperbolic fundamental group. Finally we speculate on connections between these results and a program to prove the geometrization conjecture for tautly foliated 3-manifolds.

Publication: Geometry and Topology Vol.: 4 No.: 17 ISSN: 1465-3060

ID: CaltechAUTHORS:CALgt00

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Abstract: We produce examples of taut foliations of hyperbolic 3-manifolds which are R-covered but not uniform --- ie the leaf space of the universal cover is R, but pairs of leaves are not contained in bounded neighborhoods of each other. This answers in the negative a conjecture of Thurston `Three-manifolds, foliations and circles I' (math.GT/9712268). We further show that these foliations can be chosen to be C^0 close to foliations by closed surfaces. Our construction underscores the importance of the existence of transverse regulating vector fields and cone fields for R-covered foliations. Finally, we discuss the effect of perturbing arbitrary R-covered foliations.

Publication: Geometry and Topology Vol.: 3 No.: 6 ISSN: 1465-3060

ID: CaltechAUTHORS:CALgt99

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