arXiv subjects

Chi Cheuk Tsang

Publications and source records attributed to Chi Cheuk Tsang.

18 recordsLinked to original sources

arXiv2026

Finiteness of veering triangulations

We show that a finite volume cusped hyperbolic 3-manifold admits at most finitely many veering triangulations, and in fact this number is bounded above by the number of Giroux torsion free tight contact structures. This resolves Kirby problem K3 3.21e. More generally, for any 3-manifold $M$ and any link $L$, we show finiteness for the family of pseudo-Anosov flows which admits a \emph{strictly positive Birkhoff section relative to $L$}. Combined with work of Li, this implies that a fixed closed $3$-manifold admits at most finitely many pseudo-Anosov flows without perfect fits. This resolves Kirby problem K3 3.21d.

math.GT↗

arXiv2026

Legendrian position of veering triangulations

We make a first step towards connecting the theory of veering triangulations and bicontact structures as tools for studying (pseudo-)Anosov flows: We show that given a veering triangulation corresponding to an Anosov flow with orientable stable and unstable foliations, the edges of the triangulation can be realized as Legendrian arcs with respect to a strongly adapted bicontact structure that supports the Anosov flow. Along the way, we show that every veering triangulation can be placed in `steady position', where each pair of edge projections that intersect in the orbit space only intersect once transversely. By a previous result of the author, this implies that horizontal surgery of veering triangulations correspond to horizontal Goodman surgery of pseudo-Anosov flows.

math.GT↗

arXiv2026

From pseudo-Anosov flows on graph manifolds to totally periodic flows

We show that every pseudo-Anosov flow on a graph manifold is almost equivalent, i.e. orbit equivalent in the complement of a finite collection of closed orbits, to a totally periodic pseudo-Anosov flow or a suspension Anosov flow. The proof is via a hands-on construction of a partial Birkhoff section with genus one components that misses finitely many closed orbits. When combined with previous work of the author, this implies that every transitive Anosov flow on a graph manifold with orientable stable and unstable foliations is almost equivalent to a suspension Anosov flow.

math.DS↗

arXiv2025

A "cubist" decomposition of the Handel-Mosher axis bundle

We show that the axis bundle of a nongeometric fully irreducible outer automorphism admits a canonical "cubist" decomposition into branched cubes that fit together with special combinatorics. From this structure, we locate a canonical finite collection of periodic fold lines in each axis bundle. This can be considered as an analogue of results of Hamenstädt and Agol from the surface setting, which state that the set of trivalent train tracks carrying the unstable lamination of a pseudo-Anosov map can be given the structure of a CAT(0) cube complex, and that there is a canonical periodic fold line in this cube complex. This work also gives an answer to questions of Handel-Mosher and Bridson-Vogtmann regarding the geometry of the axis bundle and a solution of a new flavor to the fully irreducible conjugacy problem in $\mathrm{Out}(F_r)$.

math.GR↗

arXiv2025

Minimum dilatations of pseudo-Anosov braids

We determine the minimum dilatation $δ_n$ among pseudo-Anosov braids with $n$ strands, for large enough values of $n$. These are the dilatations attained by the examples of Hironaka-Kin and Venzke, and they satisfy $\lim_{n \to \infty} δ_n^n = (2+\sqrt{3})^2 \approx 13.928$. Together with previous work, this result confirms conjectures by Kin-Takasawa and Venzke, and solves the minimum dilatation problem on the $n$-punctured sphere, for all but $6$ values of $n$.

math.GT↗

arXiv2025

On the set of normalized dilatations of fully-punctured pseudo-Anosov maps

We improve the bound on the number of tetrahedra in the veering triangulation of a fully-punctured pseudo-Anosov mapping torus in terms of the normalized dilatation. When the mapping torus has only one boundary component, we can improve the bound further. Together with the author's work with Hironaka in the case when the mapping torus has at least two boundary components, this allows us to understand small elements of the set $\mathcal{D}$ of normalized dilatations of fully-punctured pseudo-Anosov maps using computational means. In particular, we certify that the minimum element of $\mathcal{D}$ is $μ^2$ and the minimum accumulation point of $\mathcal{D}$ is $μ^4$, where $μ$ is the golden ratio.

math.GT↗

arXiv2025

Heegaard Floer theory and pseudo-Anosov flows II: Differential and Fried pants

In earlier work, relying on work of Agol-Guéritaud and Landry-Minsky-Taylor, we showed that given a pseudo-Anosov flow $(Y,ϕ)$ and a collection of closed orbits $\mathcal{C}$ satisfying the `no perfect fit' condition, one can construct a special Heegaard diagram for the link complement $Y^\sharp= Y \setminus ν(\mathcal{C})$ framed by the degeneracy curves. In this paper, we demonstrate how the special combinatorics of this diagram can be used to understand the differential of the associated Heegaard Floer chain complex. More specifically, we introduce a refinement of the $\text{spin}^\text{c}$-grading obstructing two Heegaard states from being connected by an effective domain. We describe explicitly the subcomplexes in the refined gradings that represent irreducible multi-orbits, in the sense that they contain states corresponding to multi-orbits which cannot be resolved along Fried pants. In particular we show that the homology of these subcomplexes are 1-dimensional. When specialized to the case of suspension flows our arguments prove some results in the spirit of Ni, Ghiggini, and Spano: the next-to-top non-zero sutured Floer group counts the number of periodic points of least period.

math.GT↗

arXiv2025

Heegaard Floer theory and pseudo-Anosov flows I: Generators and categorification of the zeta function

We bring to light a new connection between dynamics and Heegaard Floer homology. On a closed 3-manifold $Y$ we consider a pseudo-Anosov flow $ϕ$ with no perfect fits with respect to its singularity locus $L \subset Y$, or perhaps a larger collection of closed orbits. Using work of Agol and Guéritaud on veering branched surfaces we produce a chain complex computing the link Floer homology of $L$ in the framing specified by the degeneracy curves of the flow. Using work of Landry, Minsky, and Taylor we show that the generators of the chain complex correspond to certain closed multi-orbits of $ϕ$. We prove that two canonical generators $\mathbf{x}^\mathrm{top}$ and $\mathbf{x}^\mathrm{bot}$ determine non-trivial homology classes located in the $\text{spin}^\text{c}$-grading of the flow, and its opposite. Finally, we observe that our specific model of the chain complex for link Floer homology naturally supports a grading with dynamical significance. This grading, a modification of the regular Maslov grading, is shown to categorify a suitable normalization of the zeta function associated to $ϕ$.

math.GT↗

arXiv2024

Minimal stretch factors of orientation-reversing fully-punctured pseudo-Anosov maps

We show that the stretch factor $λ(f)$ of an orientation-reversing fully-punctured pseudo-Anosov map $f$ on a finite-type orientable surface $S$, with $-χ(S) \geq 4$ and having at least two puncture orbits, satisfies the inequality $λ(f)^{-χ(S)} \geq σ^2$, where $σ=1+\sqrt{2}$ is the silver ratio. We provide examples showing that this bound is asymptotically sharp. This extends previous results of Hironaka and the third author to orientation-reversing maps.

math.GT↗

arXiv2024

Endperiodic maps, splitting sequences, and branched surfaces

We strengthen the unpublished theorem of Gabai and Mosher that every depth one sutured manifold contains a very full dynamic branched surface by showing that the branched surface can be chosen to satisfy an additional property we call veering. To this end we prove that every endperiodic map admits a periodic splitting sequence of train tracks carrying its positive Handel-Miller lamination. This completes step one of Gabai-Mosher's unpublished two-step proof that every taut finite depth foliation of a compact, oriented, atoroidal 3-manifold is almost transverse to a pseudo-Anosov flow. Further, a veering branched surface in a sutured manifold is a generalization of a veering triangulation, and we extend some of the theory of veering triangulations to this setting. In particular we show that the branched surfaces we construct are unique up to a natural equivalence relation, and give an algorithmic way to compute the foliation cones of Cantwell-Conlon.

math.GT↗

arXiv2024

Examples of Anosov flows with genus one Birkhoff sections

We show that a transitive Anosov flow with orientable stable and unstable foliations that either (i) admits a Birkhoff section whose first return map is a Penner type pseudo-Anosov map, or (ii) is totally periodic admits a genus one Birkhoff section. This provides evidence for a conjecture of Fried and Ghys. The proof utilizes a result of the author on the horizontal Goodman surgery operation. To apply this result for showing (i), we establish correspondence between horizontal Goodman surgery on pseudo-Anosov flows and horizontal surgery on veering triangulations in the layered setting.

math.DS↗

arXiv2024

Veering branched surfaces, surgeries, and geodesic flows

We introduce veering branched surfaces as a dual way of studying veering triangulations. We then discuss some surgical operations on veering branched surfaces. Using these, we provide explicit constructions of some veering branched surfaces whose dual veering triangulations correspond to geodesic flows of negatively curved surfaces. We construct these veering branched surfaces on (i) complements of Montesinos links whose double branched covers are unit tangent bundles of negatively curved orbifolds, and (ii) complements of full lifts of filling geodesics in unit tangent bundles of negatively curved surfaces, when the geodesics have no triple intersections and have ($n \geq 4$)-gons as complementary regions. As an application, this provides explicit Markov partitions of geodesic flows on negatively curved surfaces. In an appendix, we classify the drilled unit tangent bundles which admit a veering triangulation corresponding to a geodesic flow, by characterizing when there are no perfect fits.

math.GT↗

arXiv2024

Standardly embedded train tracks and pseudo-Anosov maps with minimum expansion factor

We show that given a fully-punctured pseudo-Anosov map $f:S \to S$ whose punctures lie in at least two orbits under the action of $f$, the expansion factor $λ(f)$ satisfies the inequality $λ(f)^{|χ(S)|} \ge μ^4 \approx 6.85408$, where $μ= \frac{1 + \sqrt{5}}{2} \approx 1.61803$ is the golden ratio. The proof involves a study of standardly embedded train tracks, and the Thurston symplectic form defined on their weight space.

math.GT↗

arXiv2024

Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows

We provide an exposition of a `horizontal' generalization of Goodman's surgery operation on (pseudo-)Anosov flows. This operation is performed by cutting along a specific kind of annulus that is transverse to the flow and regluing with a Dehn twist of the appropriate sign. We then show that performing horizontal Goodman surgery on a transitive pseudo-Anosov flow yields an almost equivalent flow, i.e. the original flow and the surgered flow are orbit equivalent after drilling out a finite collection of closed orbits. We obtain some almost equivalence results by applying this theorem on examples of the surgery operation. Along the way, we also show a structural stability result for pseudo-Anosov flows.

math.DS↗

arXiv2024

Constructing Birkhoff sections for pseudo-Anosov flows with controlled complexity

We introduce a new method of constructing Birkhoff sections for pseudo-Anosov flows, which uses the connection between pseudo-Anosov flows and veering triangulations. This method allows for explicit constructions, as well as control over the Birkhoff section in terms of its Euler characteristic and the complexity of the boundary orbits. In particular, we show that any transitive pseudo-Anosov flow has a Birkhoff section with two boundary components.

math.DS↗

arXiv2024

Dynamics of veering triangulations: infinitesimal components of their flow graphs and applications

We study the strongly connected components of the flow graph associated to a veering triangulation, and show that the infinitesimal components must be of a certain form, which have to do with subsets of the triangulation which we call `walls'. We show two applications of this knowledge: (1) a fix of a proof in the original paper by the first author which introduced veering triangulations; and (2) an alternate proof that veering triangulations induce pseudo-Anosov flows without perfect fits, which was initially proved by Schleimer and Segerman.

math.GT↗

arXiv2022

Cohomologies on almost complex manifolds and the $\partial \bar{\partial}$-lemma

We study cohomologies on an almost complex manifold $(M, J)$, defined using the Nijenhuis-Lie derivations $\mathcal{L}_J$ and $\mathcal{L}_N$ induced from the almost complex structure $J$ and its Nijenhuis tensor $N$, regarded as vector-valued forms on $M$. We show how one of these, the $N$-cohomology $H^{\bullet}_N (M)$, can be used to distinguish non-isomorphic non-integrable almost complex structures on $M$. Another one, the $J$-cohomology $H^{\bullet}_J (M)$, is familiar in the integrable case but we extend its definition and applicability to the case of non-integrable almost complex structures. The $J$-cohomology encodes whether a complex manifold satisfies the $\partial \bar{\partial}$-lemma, and more generally in the non-integrable case the $J$-cohomology encodes whether $(M, J)$ satisfies the $\mathrm{d} \mathcal{L}_J$-lemma, which we introduce and motivate in this paper. We discuss several explicit examples in detail, including a non-integrable example. We also show that $H^k_J$ is finite-dimensional for compact integrable $(M, J)$, and use spectral sequences to establish partial results on the finite-dimensionality of $H^k_J$ in the compact non-integrable case.

math.DG↗

arXiv2018

The $\mathcal L_B$-cohomology on compact torsion-free $\mathrm{G}_2$ manifolds and an application to 'almost' formality

We study a cohomology theory $H^{\bullet}_φ$, called the $\mathcal L_B$-cohomology, on compact torsion-free $\mathrm{G}_2$-manifolds. We show that $H^k_φ \cong H^k_{\mathrm{dR}}$ for $k \neq 3, 4$, but that $H^k_φ$ is infinite-dimensional for $k = 3,4$. Nevertheless there is a canonical injection $H^k_{\mathrm{dR}} \to H^k_φ$. The $\mathcal L_B$-cohomology also satisfies a Poincaré duality induced by the Hodge star. The establishment of these results requires a delicate analysis of the interplay between the exterior derivative $\mathrm{d}$ and the derivation $\mathcal L_B$, and uses both Hodge theory and the special properties of $\mathrm{G}_2$-structures in an essential way. As an application of our results, we prove that compact torsion-free $\mathrm{G}_2$-manifolds are 'almost formal' in the sense that most of the Massey triple products necessarily must vanish.

math.DG↗