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3:30-4:00 |
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Tea |
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4:00-4:50 |
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Colloquium: Ruth Charney |
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From Artin monoids to Artin groups
Slides |
5:00-6:00 |
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Lightning talk session 1 |
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8 talks see below
Slides
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6:30- |
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Social Gathering |
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Club Room at the Georgia Tech Hotel
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9:00-9:30 |
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Bagels and light refreshments |
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9:30-10:20 |
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Talk: Dave Gabai |
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Dax's work on embedding spaces
Scanned Notes |
10:30-11:00 |
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Tea Break |
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11:00-11:30 |
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Talk: Abdoul Karim Sane |
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The Thurston norm and a baby version: the intersection norm
Scanned Notes |
11:30-12:00 |
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Talk: Wade Bloomquist |
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Congruence subgroups of the braid group
Scanned Notes |
12:00-12:45 |
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Catered Lunch |
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Fox Bros BBQ |
12:45-1:30 |
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Asynchronous lightning talk discussion session |
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6 speakers see below
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1:30-2:00 |
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break |
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2:00-2:50 |
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Talk: Yvon Verberne |
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Automorphisms of the fine curve graph
Scanned Notes |
3:00-4:00 |
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Extended Tea Break |
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4:00-4:50 |
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Talk: Hannah Schwartz |
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Gluck Twisting 2-Spheres
Slides |
6:00- |
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Banquet |
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At: South City Kitchen
and Lure |
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9:00-9:30 |
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Light refreshments |
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9:30-10:00 |
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Talk: Jonathan Simone |
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Links and rational homology 4-balls
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10:00-10:30 |
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Talk: Hannah Turner |
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Generalizing the (fractional) Dehn twist coefficient
Slides
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10:30-11:30 |
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Lighting talk session 2 |
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7 talks see below
Slides
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11:30-12:00 |
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Tea Break |
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12:00-12:50 |
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Talk: Maggie Miller |
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Concordance of surfaces
Slides |
| | Christian Millichap | Flat fully augmented links are determined by their complements |
| | James Hughes | Legendrian loops and mapping class groups |
| | Deepisha Solanki | Plat Representations of the Unknot |
| | Jacob Russell | Purely pseudo-Anosov subgroups of fibered 3-manifold groups |
| | Lizzie Buchanan | A new condition on the Jones polynomial of a fibered positive link |
| | Alex Nolte | Higher complex structures and the SL(3,R) Hitchin component |
| | Joshua Perlmutter | Extending Group Actions on Metric Spaces |
| | Melody Molander | Diagrammatic Presentations of Index 4 Subfactor Planar Algebras |
| | Assaf Bar-Natan | How the Thurston Metric on Teichmuller Space is (not) Like L^(infty) |
| | Vivian He | Boundaries at Infinity |
| | Nicholas Wawrykow | Disk Configuration Spaces and Representation Stability |
| | Eduardo Fernández Fuertes | The Legendrian Unknot in a tight contact 3-manifold |
| | Ethan Dlugie | The Burau Representation and Shapes of Polyhedra |
| | Ryan Stees | Milnor’s invariants for knots and links in closed orientable 3-manifolds |
| | Roberta Shapiro | Automorphisms of the fine 1-curve graph |
| | Hong Chang | Efficient geodesics in the curve complex and their dot graphs |
| | Vo Hahn | Short closed geodesics with self-intersections |
| | Vijay Higgins | Stated skein algebras for Kuperberg webs |
| | Oguz Savk | On homology planes and contractible 4-manifolds |
| | Shunyu Wan | Naturality of Legendrian LOSS invariant under positive contact surgery. |
| | Bojun Zhao | Left orderability and taut foliations with one-sided branching |
Titles and Abstracts
Congruence subgroups of the braid group
The integral Burau representation provides a homomorphism from the braid group to matrices with integer entries. Reducing these entries modulo m allows for an analog of congruence subgroups to be defined for braid groups. These finite index subgroups, called level m braid groups, will be discussed with a focus on their images under the integral Burau representation. We will conclude with a brief discussion of the Property F conjecture and how the above example might serve as a stepping stone for providing some insight into a much wider class of braid group representations. This is joint work with Nancy Scherich and Peter Patzt.
From Artin monoids to Artin groups
Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form and have played a major role in geometric group theory and low-dimensional topology in recent years. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges. However, if one restricts to the Artin monoid, then much of the combinatorial structure still applies in the infinite-type case. In a joint project with Rachael Boyd, Rose Morris-Wright, and Sarah Rees, we use geometric techniques to study the relation between the Artin monoid and the Artin group.
Dax's work on embedding spaces
More than 50 years ago Jean Pierre Dax developed an abstract theory of homotopy groups of embeddings of one manifold into another subject to certain conditions. We will discuss the simplest case of Dax's work, that of loops of proper embeddings of the interval into a 4-manifold, that are contractible in the space of maps of the interval fixing the boundary. It is a strikingly clean and beautiful geometric result with an elementary proof. We will use it to show that a particular pair of embedded discs in a compact 4-manifold M are not isotopic rel boundary even though there is a diffeomorphism of M homotopic to id, fixing the boundary pointwise, taking one disc to the other.
Concordance of surfaces
Topologists commonly study concordance of knots in 3-manifolds, and in the past few decades have developed increasingly sophisticated ways to distinguish knots up to concordance. While in seemingly analogous, concordance of surfaces in 4-manifolds actually behaves extremely differently: for example, Kervaire showed that every 2-sphere in S^4 is concordant to the unknotted 2-sphere. I’ll discuss these definitions, distinctions, and how one can study concordance of 2-spheres in other 4-manifolds. My part of the work discussed is joint with Michael Klug.
The Thurston norm and a baby version: the intersection norm
The Thurston norm on a 3-manifold M is a norm on the second homology of M. Introduced by W. Thurston, it is an invariant of 3-manifolds that encodes important features like the existence of foliations without Reeb components. It is a result of Thurston that certain points inside the dual unit ball of the norm on a 3-manifold M correspond to Euler classes of foliations without Reeb components on M. Dual unit balls of Thurston norms happen to be symmetric polytopes and it is general problem to characterize polytopes that appear as the dual unit ball of a Thurston norm. In this talk, we will explain an analogy between Thurston norms on 3-manifolds and another family of norms, called the intersection norms, defined on the first homology of a closed oriented surface. More than a simple analogy, we will show that these two norms coincide in some cases and we will use this fact to provide a large family of polytopes that are Thurston balls.
Gluck Twisting 2-Spheres
The smooth, generalized Poincar\'e conjecture -- that all smooth homotopy spheres are diffeomorphic to the standard one -- remains unknown only in dimension four. In this talk, we will examine homotopy 4-spheres called ``Gluck twists" obtained by the operation of ``Gluck twisting" embedded 2-spheres in $S^4$. Although this operation was defined by Gluck in the 60's, many Gluck twists are still not known to be standard. We will discuss a variety of distinct viewpoints from which to understand the diffeomorphism type of Gluck twists, including joint work with several different combinations of her collaborators Gabai, Naylor, Joseph, Ruppik, and Klug.
Links and rational homology 4-balls
It is well-known that the double cover of the 3-sphere branched along a slice knot bounds a rational homology 4-ball. There is a generalization of sliceness to links that preserves this property; a result of Donald-Owens shows that if a nonzero determinant link L bounds a smooth, properly embedded surface in the 4-ball with no closed components and Euler characteristic 1, then the double cover of the 3-sphere branched along L bounds a rational homology 4-ball. Such links are called chi-slice. In this talk, I will discuss the chi-sliceness of pretzel links and 3-braid links, obstructions to chi-sliceness, and why chi-slice links and the classical notion of slice links are both natural generalizations of the notion of slice knots.
Generalizing the (fractional) Dehn twist coefficient
The fractional Dehn twist coefficient (FDTC) is a rational number associated to a mapping class on a (finite-type) surface with boundary. This 2-dimensional invariant has many applications to 3-manifold topology and contact geometry. One way to think of the FDTC is as a real-valued function on the mapping class group of a surface with many nice properties. In this talk, we will give sufficient conditions on a more general group to admit a function which behaves like the FDTC. In particular, we use this to generalize the FDTC to infinite-type surfaces (with boundary); in this setting, we show that the "fractional" Dehn twist coefficient need not be rational. This is joint work in progress with Peter Feller and Diana Hubbard.
Automorphisms of the fine curve graph
The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.
Lightning talk
How the Thurston Metric on Teichmuller Space is (not) Like L^(infty)
The Thurston Metric, introduced by Thurston in 1986, is an asymmetric metric on Teichmuller space, which measures distance between surfaces using the Lipschitz constant of maps between them. In this talk, I will tell you what I know about geodesics in this metric. Specifically, I will tell you about the geodesic envelope, its shape (and how the Thurston metric is similar to L^(infty)), and its width (and how the Thurston metric is not similar to L^(infty)). I'll state a theorem which gives sufficient conditions for geodesics between two points to be "essentially unique" (ie, uniformly bounded diameter from each other).
A new condition on the Jones polynomial of a fibered positive link
The maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree, and we have a similar result for links. The key to our proof is the notion of a Balanced diagram and its special properties. This theorem allows us to complete the positivity classification of all knots up to 12 crossings. This talk is based on arXiv:2204.03846.
Efficient geodesics in the curve complex and their dot graphs
The notion of efficient geodesics in $\mathcal{C}(S_{g>1})$, the complex of curves of a closed orientable surface of genus $g$, was first introduced in "Efficient geodesics and an effective algorithm for distance in the complex of curves". There it was established that there exists (finitely many) efficient geodesics between any two vertices, $ v_{\alpha} , v_{\beta} \in \mathcal{C}(S_g)$, representing homotopy classes of simple closed curves, $\alpha , \beta \subset S_g$. The main tool for used in establishing the existence of efficient geodesic was a {\em dot graph}, a booking scheme for recording the intersection pattern of a {\em reference arc}, $\gamma \subset S_g$, with the simple closed curves associated with the vertices of geodesic path in the zero skeleton, $\mathcal{C}^0(S_g)$. In particular, it was shown that any curve corresponding to the vertex that is distance one from $v_\alpha$ in an efficient geodesic intersects any $\gamma$ at most $d -2$ times, when the distance between $v_\alpha$ and $v_\beta$ is $d \geq 3$. In this note we make a more expansive study of the characterizing ``shape'' of the dot graphs over the entire set of vertices in an efficient geodesic edge-path. The key take away of this study is that the shape of a dot graph for any efficient geodesic is contained within a {\em spindle shape} region.
The Burau Representation and Shapes of Polyhedra
We still don't know if the Burau representation of the 4-strand braid group is a faithful representation. Many topologists should be interested in this question, for instance because a negative answer would indicate a knot with trivial Jones polynomial. By highlighting a connection between the Burau representation and a moduli space of flat cone metrics on spheres originally explored by W. Thurston, we can put strong restrictions on the kernel of the $n=4$ Burau representation.
The Legendrian Unknot in a tight contact 3-manifold
The path-connected components of the space of Legendrian unknots in a tight contact 3-manifold were classified by Eliashberg and Fraser in 1998. They showed that two Legendrian unknots in a tight contact 3-manifold are Legendrian isotopic if they have the same Thurston-Bennequin invariant (tb) and rotation number (rot). Moreover, they determined the combinations (tb,rot) that are realized by some Legendrian unknot in a tight contact 3-manifold. In this talk I will prove that the space of parametrized ''long'' Legendrian unknots in a tight contact 3-manifold with (tb,|rot|)=(tb,-1-tb) is homotopy equivalent to the space of parametrized smooth long unknots. This is joint work with J. Martínez-Aguinaga and F. Presas.
Short closed geodesics with self-intersections
We consider the set of closed geodesics on cusped hyperbolic surfaces. Given any non-negative integer k, we are interested in the set of closed geodesics with at least k self-intersections. Among these, we investigate those of minimal length. In this talk, we will discuss their self-intersection numbers.
Boundaries at Infinity
The central idea of geometric group theory is to study groups via their actions on topological spaces. One object of interest is the boundary at infinity. A classical example is the Gromov boundary of a hyperbolic space exhibiting many nice properties: it is compact, metrizable, invariant under quasi-isometries, and witnesses random walks. This motivates the problem of generalizing the Gromov boundary to general geodesic metric spaces, that is, to define a topological boundary satisfying some of these properties. In this talk, we will discuss some of these generalizations, including the contracting boundary, the Morse boundary and the sublinearly Morse boundary.
Stated skein algebras for Kuperberg webs
The skein algebra of a surface is spanned by links in the thickened surface subject to skein relations. A finer version of the skein algebra, called the stated skein algebra, was introduced by Thang Le and is compatible with the cutting and gluing of surfaces. The stated skein algebra gives a diagrammatic way to encode the quantum group associated to the skein relations. In this lightning talk we will explore diagrammatic bases for these stated skein algebras when the skein relations are Kuperberg's web relations associated to lie algebras of rank 2.
Legendrian loops and mapping class groups
Many recent advances in the classification of exact Lagrangian fillings of Legendrian links come from the study of Legendrian loops, i.e. Legendrian isotopies pointwise fixing a given Legendrian link. Legendrian loops induce an (often nontrivial) action on sheaf-theoretic or Floer-theoretic invariants of the Legendrian. In this talk, I will highlight some results about groups of Legendrian loops and briefly hint at how cluster-algebraic structures on the moduli of sheaves or the augmentation variety connect Legendrian loop actions to the theory of mapping class groups.
Non-standard orders on torus bundles with one boundary
Consider a torus bundle over the circle with one boundary. Perron-Rolfsen shows that having an Alexander polynomial with real positive roots is a sufficient condition for the bundle to have bi-orderable fundamental group. This is done by showing the action of the monodromy induced on the fundamental group of the surface preserves a “standard” bi-ordering. In this talk, we discuss non-standard bi-orderings of the fundamental group of a torus with one boundary preserved by the monodromy of a torus bundle. This work is joint with Henry Segerman. This work is partially funded by the NSF Ascend Fellowship (DMS-2213213).
Flat fully augmented links are determined by their complements
The Gordon-Luecke Theorem states that knots are determined by their complements, that is, if two knot complements are homeomorphic, then these knots are equivalent. While this fact does not hold when we expand to links, it is natural to ask: are certain infinite classes of links determined by their complements? In this talk, we will introduce a large and interesting class of links, called flat fully augmented links, and briefly discuss how the hyperbolic geometry of such links can be exploited to show that they are determined by their complements.
Diagrammatic Presentations of Index 4 Subfactor Planar Algebras
Subfactor planar algebras first appeared out of studying the standard invariant of a subfactor. Planar algebras can be conveniently encoded by diagrams in the plane. These diagrams satisfy some skein relations and have an invariant called an index. The Kuperberg Program asks to find all diagrammatic presentations of subfactor planar algebras. This program has been completed for index less than 4. In this talk, I will introduce subfactor planar algebras and give presentations for subfactor planar algebras of index 4.
Higher complex structures and the SL(3,R) Hitchin component
I'll discuss higher complex structures, analogues of complex structures on surfaces that are conjectured to parametrize PSL(n,R) Hitchin components. I'll discuss recent results on the structure of the moduli space of degree n-complex structures, and a solution to the first nontrivial case of the main conjecture on higher complex structures.
Extending Group Actions on Metric Spaces
Geometric group theory involves the study of groups via their actions on metric spaces. If a subgroup acts on a metric space, it is natural to ask if such an action can be extended to an action of the whole group on a (possibly different) metric space. It is known that if H is hyperbolically embedded in G then any isometric action of H on a metric space can be extended. I am interested in generalizing this result to non-hyperbolically embedded subgroups, by instead putting restrictions on the group actions.
Purely pseudo-Anosov subgroups of fibered 3-manifold groups
Farb and Mosher's convex cocompact subgroups are some of the geometrically, dynamically, and algebraically richest subgroups of the mapping class group. A major open question about these subgroups asks if they are characterized by each element acting with pseudo-Anosov dynamics on the surface. We show the answer is 'yes' when you restrict to subgroups of fibered 3-manifold groups included into the mapping class group via the Birman exact sequence.
On homology planes and contractible 4-manifolds
We provide the first additional examples of homology spheres that bound both homology planes and Mazur or Poénaru manifolds. Before that there was a single example due to Kirby and Ramanujam; therefore, we call such homology spheres Kirby-Ramanujam spheres. Also, we show that one of our families of Kirby-Ramanujam spheres is diffeomorphic to the splice of two certain families of Brieskorn spheres. Even though our new examples lie in the class of the trivial element in the homology cobordism group, both splice components are separately linearly independent in that group. This is joint work with Rodolfo A. Aguilar.
Automorphisms of the fine 1-curve graph
The fine curve graph of a surface S was introduced by Bowden–Hensel–Webb in 2019 to study the diffeomorphism group of S. We build on the works of Long–Margalit–Pham–Verberne–Yao and Le Roux–Wolff to show that the automorphism group of the fine 1-curve graph, a variant of the fine curve graph, is also the homeomorphism group of S. Joint work with K. W. Booth and D. Minahan.
Plat Representations of the Unknot
The main result is a version of Birman's theorem about equivalence of plats, which does not involve stabilisation, for the unknot. We introduce the 'flip move', which modifies a plat without changing its link type or bridge index. Our main result shows that the flip move is the only obstruction to reducing a closed $2n$- plat representative of the unlink to the standard 0-crossing unlink, through a sequence of plats of non-increasing bridge index.
Milnor’s invariants for knots and links in closed orientable 3-manifolds
Early in his career, John Milnor defined his seminal link invariants, now called Milnor's $\overline{\mu}$-invariants. They are concordance invariants of links in $S^3$, and much is known about them. I will discuss an extension of these invariants to concordance invariants of knots and links in any closed orientable 3-manifold, state some theorems which justify calling them ``Milnor’s invariants", and connect them to previous work.
Naturality of Legendrian LOSS invariant under positive contact surgery.
Given a Legedrian Knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L qwhich lives in the knot Floer homology group. We proved that the LOSS invariant is natural under the positive contact surgery, and use it to distinguish Legendrian and transverse knot.
Disk Configuration Spaces and Representation Stability
Given a manifold $X$ with metric $g$, consider the ordered configuration space of $n$ unit-diameter disks in $(X, g)$. The $n^{\text{th}}$-symmetric group acts on the homology of this configuration space; by fixing homological degree $k$ and letting $n$ vary, we get a sequence of symmetric group representations. In this talk, we will see that if $(X,g)$ is the infinite Euclidean strip of width at least $2$, then this sequence of symmetric group representations stabilizes in a reasonable sense.
Left orderability and taut foliations with one-sided branching
Let M be a closed orientable irreducible 3-manifold that admits a co-orientable taut foliation F. We provide some results to show that π1(M) is left orderable in the following cases: (1) Suppose that M admits a co-orientable taut foliation with one-sided branching, then π1(M) is left orderable. (2) Suppose that M admits a co-orientable taut foliation with orderable cataclysm, then π1(M) is left orderable. We give some examples of taut foliations with this property: 2-a: If an Anosov flow has co-orientable stable and unstable foliations, then the stable and unstable foliations have orderable cataclysm. In this case, it’s known that π1(M) is left orderable by the works of Thurston, Calegari-Dunfield, Boyer-Hu and Boyer-Rolfsen-Wiest. Our result gives a new proof, and the left-invariant order of π1(M) comes from a different way. 2-b: Assume that a pseudo-Anosov flow has co-orientable stable and unstable singular foliations, and the stabilizer at every singular orbit does not rotate the prongs, then the resulting foliation obtained from splitting the stable singular foliation and filling with monkey saddles has orderable cataclysm.
