Papers

We develop an algorithm that takes as input a veering triangulation V and a flow cycle c of V, and outputs a veering parent V_c of V. Constructing veering parents combinatorializes drilling out closed orbits of transitive pseudo-Anosov flows. The algorithm relies on locally approximating the structure of the loom space associated to V in sufficient detail to carry out a certain modification of the Agol-Guéritaud construction. We use an implementation of the drilling algorithm to construct layered parents of non-layered veering triangulations (and hence Birkhoff sections for transitive pseudo-Anosov flows) and geometric parents of non-geometric veering triangulations. We also find closed orbits of (drilled) pseudo-Anosov flows that are not ambiently isotopic to the geodesics in their free homotopy classes, and show that certain veering triangulations encode almost orbit equivalent flows. In particular, we provide new computational evidence for a conjecture of Ghys asserting that all transitive Anosov flows with orientable invariant foliations are almost orbit equivalent.

Let $\Gamma<\operatorname{PSL}_2(\mathbb R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $\rho:\Gamma\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen–Margulis–Sullivan measures on $\rho(\Gamma)\backslash G/M$, where $M$ is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every $u$ in the interior of the limit cone of $\rho(\Gamma)$, correlations of smooth compactly supported functions along $\exp(tu)$, normalized by $t^{(\operatorname{rank}_{\mathbb R}G-1)/2}$, admit a complete asymptotic expansion in integer powers of $t^{-1}$. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on $L^2(\rho(\Gamma)\backslash G)$ and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of $\Gamma$ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of $\rho(\Gamma)$ in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.

We study arrangements of conics and lines in the complex projective plane from the viewpoint of 4-manifolds. We introduce combinatorial conic-line arrangements, the conic-line counterpart of rank-3 matroids, together with their topological and smooth realisations, and we ask which restrictions on a complex arrangement survive when its components are replaced by locally-flat spheres in the same homology classes with the same local singularity models. For an ADE combinatorial conic-line arrangement of even total degree admitting a topological realisation we prove a Hirzebruch-type inequality; the proof uses only branched covers and the local topology of rational double points. In the complex-algebraic category we show that equality holds exactly when the arrangement is a maximising curve, hence free with prescribed exponents. We then introduce divisible and odd combinatorial conic-line arrangements and obtain inequalities and congruences for the multiplicity sequence of their resolutions, via cyclic branched covers, the G-signature theorem, spin structures, and Furuta's 10/8-theorem.

Using skein-theoretic methods, we study the SL2(C) character variety of the five-punctured sphere with arbitrary conjugacy class at each of the five punctures. We show that a Fricke-Klein-Vogt-type relation is realized as a symmetric determinantal hypersurface. We also give a realization of the character variety in terms of cluster variables. We explicitly derive the Poisson structure of the simple closed curves on the surface, and prove that cluster mutations induce automorphisms of the character variety.

Given a finite-type orientable surface without boundary, outside of a few low-complexity surfaces, we show that any measurable embedding of the associated mapping class group into itself is a measure equivalence. This is a measure-theoretic analog to the result of Ivanov and McCarthy that mapping class groups are co-Hopfian.

We show that every triple of integers $(a,b,c)$ satisfying $1\leq a\leq b\leq c$ is realized as the smooth slice genus, smooth concordance genus, and Seifert genus of a knot, respectively. This proves a conjecture of Kearney.

Let $M$ be a compact connected oriented smooth manifold, of dimension $n\ge2$. Let $\mathcal T_MG$ be one of the following transformation groups: $Homeo_0(M,\mu)$, $Diff_0(M,\mu)$, $Symp_0(M, \omega)$ (in case $M$ is symplectic) and $Ham(M, \omega )$ (in case $M$ is symplectic). Denote by $\overline{H}_b^d(\mathcal T_M)$ reduced bounded cohomology of $\mathcal T_M$ in degree $d$, and by $\overline{EH}^d(\mathcal T_M)$ reduced exact bounded cohomology of $\mathcal T_M$ in degree $d$. In this paper we prove that if $n=2$ and $M$ is a closed surface $\Sigma_g$ or a disc $\mathbb D$, then for every $d\ge 2$ $$\dim(\overline{H}b^d(\mathcal T_M))=\infty.$$ Moreover, $\dim\overline{EH}^d(\mathcal T_M)=\infty$ if $\mathcal T_M$ is either $Diff_0(\Sigma_g,\mu)$ or $Ham(\Sigma_g, \omega)$, or $Ham(\mathbb{D}, \omega)$; or $g>1$. In case $n>2$, under certain conditions on $\pi_1(M)$, we prove that for every degree $d\ge 2$ $$\dim\overline{EH}^d(\mathcal T_M)=\infty.$$ In particular, these results hold for $\mathcal T_M$ when $M$ is a closed, orientable smooth manifold admitting a Riemannian metric of strictly negative sectional curvature of an arbitrary dimension.

We develop a theory of bi-contact plugs on oriented $3$-manifolds to construct Anosov flows or, more generally, structurally stable nonsingular flows. For a hyperbolic plug with filling Morse-Smale boundary laminations, we prove that every strongly transverse gluing map can be isotoped through strongly transverse maps to identify a strongly adapted unoriented contact form. The resulting flow is hyperbolic and is Anosov when the quotient is closed. This answers a question of Béguin-Bonatti-Yu. The result also extends to gluing nonsingular partially hyperbolic flows. Anosov completion by reflection embeds each such hyperbolic plug in an Anosov flow on its oriented double. Further applications include a classification of structurally generic projectively Anosov flows without saddle periodic orbits and Morse-Smale examples with prescribed saddle count. Generalizing the construction of Bonatti-Bowden-Potrie, finite capping embeds any bi-contact plug in a closed projectively Anosov flow with the same topological entropy, preserving partial hyperbolicity for attracting plugs. We also construct flows with controlled invariant torus dynamics on torus bundles and on doubles of plugs with coorientable boundary bi-foliations. Finally, every cooriented partially hyperbolic bi-contact plug admits a Liouville structure on its thickening, with Liouville trajectories projecting to positively reparametrized flow lines. Repelling plugs yield four-dimensional Liouville domains, often with chaotic skeletons.

Let $M$ be a closed, smooth or topological $n$-manifold, with $n \geq 4$. We construct a homotopy highly cartesian square relating the space ${\mathcal E}(M (\ell))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $\ell$-sections of its stable normal microbundle, and an $(\infty + n)$-fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of ${\mathcal E}(M(\ell))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton–Kreck for closed, oriented $4$-manifolds.

We develop a theory of ending currents for properly discontinuous isometric actions of hyperbolic groups on Gromov hyperbolic spaces, generalizing Thurston's ending laminations. Several natural generalizations are considered, which agree in the 3-dimensional hyperbolic setting but are otherwise distinct. We illustrate these invariants and their differences by several examples, including closed surface group actions on hyperbolic group extensions arising from Teichmüller theory and a geometrically finite action of a closed surface group on $\mathbb{H}^4$ with a single cusp that corresponds to a non-simple curve on the surface. We also discuss the conditions under which ending currents characterize geometrically finite actions, obtaining several equivalent conditions and some surprises, illustrated by an example of a geometrically infinite action whose ending currents are all supported on a single orbit of peripheral subgroups.

We introduce, for an embedded Legendrian link~$L$ in a contact 3-manifold~$(M,\xi)$, the group of its Legendrian motions. Analogously with the smooth case, we prove that this Legendrian motion group is isomorphic to the fundamental group of the space of unparametrised Legendrian embeddings. For all Legendrian knots in the standard contact 3-sphere, we characterise Legendrian motions in terms of contact mapping classes of the knot exterior. In an appendix, we give a short exposition of our techniques applied to the smooth case, comparing to the classical Dahm homomorphism.

Let $\mathop{Mod}_{g,n}$ be the mapping class group of a genus-$g$ surface with $n$ punctures and let $\mathcal{C}_{g,n}$ be its curve complex. For a finite-index subgroup $G < \mathop{Mod}_{g,n}$, Boggi proved that $\widetilde{H}_k(\mathcal{C}_{g,n}/G;\mathbb{Q}) = 0$ for $g+n \gg k$. This plays an important role in the work of Putman-Wieland on the virtual first Betti number of $\mathop{Mod}_{g,n}$. The proof of Boggi's theorem uses mixed Hodge theory and is embedded in his flawed paper purporting to show that the mapping class group has the congruence subgroup property. We give a detailed exposition of Boggi's proof aimed at geometric topologists.

We show that, given a closed connected non-orientable surface $F$ and a simple branched cover $p \colon F \rightarrow \mathbb{RP}^2$ of degree $d \geq 5$, every self-diffeomorphism of $F$ is isotopic to the lift of a self-diffeomorphism of $\mathbb{RP}^2$ with respect to $p$. As an application, we show that every Lefschetz fibration over a connected surface $B$ with $\partial B\neq \emptyset$ and closed non-orientable regular fiber can be expressed as the composition of a simple cover of $B\times \mathbb{RP}^2$ branched over a surface braided over $B$ and projection onto the $B$-factor.

We study Legendrian realizations of double twist knots in the standard tight contact 3-sphere. For the family K(4,m), we give upper bounds on the number of oriented and unoriented Legendrian isotopy classes of maximal Thurston–Bennequin representatives. The proof uses bypass techniques in convex surface theory and the classification of tight contact structures.

We show that every orientation-preserving action of a finite group on a closed nonpositively curved 3-manifold preserves a nonpositively curved metric.

Let $\boldsymbol{n}=(n_1,n_2,n_3):(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\boldsymbol{n})_0=0$, then $\mathrm{ord}_0(\boldsymbol{n})\in\{2,3,4\}$. This reduces Lê's conjecture to excluding potential counterexamples $\boldsymbol{n}$ of orders $2$, $3$, and $4$. The proof combines techniques from the theory of plane curve singularities, explicit cobordism constructions, and genus bounds obtained by applying properties of the $\Upsilon$ invariant coming from knot Floer homology.

We show that zero is a characterising slope for the untwisted Whitehead double of any knot $K$. More generally, this holds for all $n$-twisted Whitehead doubles of $K$, with the assumption that $n\ne pq$ if $K$ is a non-trivial $(p,q)$-torus knot or $(p,q)$-cable. In particular, zero is a characterising slope for all twist knots.

The Melvin–Morton–Rozansky theorem relates the large-colour semiclassical limit of the coloured Jones polynomial to the inverse Alexander polynomial. Bar-Natan and Garoufalidis [BNG96] proved the theorem using weight systems and, more generally, established the corresponding formula for arbitrary complex semisimple Lie algebras. We give a direct proof of this semisimple Lie algebra generalization by analysing a state sum. For a complex semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$, we use the embedding $V_{d\lambda}\hookrightarrow V_{\lambda}^{\otimes d}$ to analyse the state sum. Under the specialization $q=e^{h/d}$, we show that, as $d\to\infty$, only the identity terms and those proportional to $E_{\alpha}\otimes F_{\alpha}$ contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the $\mathfrak{sl}_{2}$ case. Writing $t_{\alpha}=e^{-2(\lambda,\alpha)h}$, the limit is therefore given by the product of $\Delta_{K}(t_{\alpha})^{-1}$ over the positive roots $\alpha$.

Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant $\unicode{x0414}(T)$ associated to a $4$-ended oriented tangle $T$ extends to a twisted complex $\unicode{x0414}^\infty(T)$ over an $A_\infty$-enhancement of the Bar-Natan algebra. We prove that $\unicode{x0414}^\infty(T)$ itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a matrix multifactorization $\mathcal M(T)$, generalizing a construction of Ballinger to the coefficient ring $\mathbb Z[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.

Links are disjoint unions of circles smoothly embedded in $S^3$. We use reinforcement learning and Bayesian optimisation to obtain new upper bounds on several link invariants that are not known to be algorithmically computable: the slice genus and the unknotting number for links, and the strong slice genus for algebraically split links. We also compute lower bounds using known invariants. Combining the upper and lower bounds, we obtain new exact values in many cases. Our unknotting agents can reproduce the non-additivity of the unknotting number for several counterexamples due to Brittenham and Hermiller, in some cases finding new unknotting trajectories.

A negative answer to Whitehead's asphericity problem would follow from an infinite chain of two-complexes, starting with a non-aspherical one, in which every inclusion induces the zero map on second homotopy groups. We prove that a connected two-complex $K$ is the first term of such chains of every finite length if and only if $K$ has a connected acyclic regular cover. In particular, there is a finite non-aspherical two-complex $K$, a presentation complex of $\mathrm{SL}(2,5)$, such that for every $n$ there is a strictly increasing chain of finite two-complexes $K=K_0\subset K_1\subset\dots\subset K_n$ in which all inclusions are zero on $\pi_2$.

We show that a knot in the 3-sphere with a nontrivial Jones polynomial of span at most 14 admits no purely cosmetic surgery. This extends a result of the first author for span at most 11. We determine all Laurent polynomials of span 15 satisfying the conditions on the Jones polynomial used in the proof. These polynomials form an infinite family. We also show that a knot of braid index at most 4 whose Jones polynomial has odd span admits no purely cosmetic surgery.

Let $A$ be the fixed point set of a semifree action of a compact (metric) group $G$ on a Keller space $X$ (an infinite-dimensional compact convex subset of a separable Fréchet space). All Keller spaces are homeomorphic and all nontrivial compact Lie groups act on all $X$'s semifreely with unique fixed points (see Section 3). We prove (1) if $A$ is a single point, then it is the fixed point set of a semifree action of $G$ with uniformly small orbits, (2) if $A$ has Property Z (e.g., is of infinite codimension) in $X$, then the action of $G$ may be replaced by one with fixed point set $A$ and uniformly small orbits, and (3) if $A$ is an \ANR, then $G$ acts on $X$ semifreely with fixed point set homeomorphic to any compact (metric) space that is Shape equivalent to $A$. Corollaries are (a) every Cell-like set (i.e., continuum of trivial Shape) embedded in a Keller space $X$ as a Z-set is the fixed point set of semifree actions on $X$ with uniformly small orbits of all nontrivial compact Lie groups (Corollary \ref{CE}) and (b) (a partial converse to a Theorem of P.\ A.\ Smith) if a compact (metric) space $Y$ has the Shape of a compact \ANR, then $Y$ embeds in $X$ as the fixed point set of a homeomorphism of $X$ of period $n$ if it is acyclic in \v Cech homology with $\mathbb{Z}_{n}$ coefficients. Finally, we (4) apply recent results of Cappell, Weinberger, and Yan \cite{cwy} to give for a compact manifold $M$ modeled on a Keller space homological and algebraic K-theoretic conditions equivalent to the existence of semifree actions of finite groups $G$ on $M$ with prescribed \ANR\ fixed point sets and (5) generalize (3) to prove that if $A$ is an \ANR\ that is the fixed point set of a semifree action of $G$ on $M$, then every compact (metric) space that is Shape equivalent to $A$ is homeomorphic to the fixed point set of a semifree action of $G$ on $M$.

$\mathbb{Z}_p$-folded surfaces are a class of oriented folded surfaces with a fundamental $H_2(\mathbb{Z}/p\mathbb{Z})$ class. In this paper we study the $\mathbb{Z}_p$-folded surfaces that embed in $3$-manifolds. This perspective naturally leads to the definition of the $\mathbb{Z}_p$-Thurston norm, which generalises the $\mathbb{Z}_2$-Thurston norm of Jaco, Rubinstein and Tillmann and adapts Turaev's definition of $\theta$. We define canonical $\mathbb{Z}_p$-folded surfaces, which connect the theory of $\mathbb{Z}_p$-folded surfaces to the study of one-vertex $3$-manifold triangulations, and we relate their Euler characteristic to the $\mathbb{Z}_p$-Thurston norm. We generalise a classical result of Bredon and Wood by proving that the maximum Euler characteristic of a non-separating non-empty $\mathbb{Z}_3$-folded surface embedded in $L(3n,1)$ is $2-n$ for $n\in\mathbb{Z}^{>0}$, determining the $\mathbb{Z}_3$-Thurston norm in these cases. We conclude by demonstrating the motivating application of this theory to the study of generalised triangulations by proving that any generalised triangulation of the lens space $L(3n,1)$ must have at least $n$ tetrahedra. This final result is a preview of joint work with Spreer in which we determine the exact complexity of $L(3n,1)$ using more sophisticated techniques.

We prove that the Laurent polynomial in $\mathbb{Z}[q^{\pm 1}]$ that is the top coefficient of the Links-Gould invariant of the boundary of a Seifert surface is multiplicative under arbitrary Murasugi sums of minimal genus Seifert surfaces. This extends a previous result of the second and third authors which only holds for plumbing, i.e. 4-Murasugi sum. Our technique relies on quantum supergroup themselves and PBW basis rather than the skein relations used in the previous work.

We unify Jones' representations of braid groups and the oriented Thompson's group by introducing the oriented dyadic braid group. This finitely generated group contains every braid group and the oriented Thompson's group. We construct an embedding of the oriented Thompson's group into this group such that taking braid closures recovers Jones' construction of oriented links. We also introduce the dyadic Temperley-Lieb algebra and construct a unitary representation of the oriented dyadic braid group extending the classical Jones representations of braid groups. Along the Thompson group's embedding, its distinguished matrix coefficient agrees with the corresponding coefficient of Jones' representation of the oriented Thompson's group.

We study reconstruction of graded surfaces with stops $(S,M,\eta)$ from their mapping class groups and the derived Picard groups of their partially wrapped Fukaya categories. In genus at least two, we prove that the ungraded mapping class group determines the surface with stops $(S,M)$, while the derived Picard group with the shift distinguished determines the underlying graded surface $(S,\eta)$. Nevertheless, the derived Picard group with both the shift and the Serre functor distinguished has countably infinite fibres. We also construct arbitrarily large finite families of pairwise non-equivalent categories with the same invariant and Grothendieck rank. For line fields induced by nowhere-vanishing vector fields, we show that the rational Serre representation determines the surface with stops $(S,M)$ in every genus. For arbitrary line fields in genus at least two, the full integral derived Picard representation, with the shift distinguished, has fibres of cardinality at most four. It completely determines the graded surface with stops $(S,M,\eta)$ when $S$ has one or at least five boundary components, while non-singleton fibres occur for two, three, and four boundary components.

We study colored knot Floer homology via a bordered cabling construction. We construct a cabling module and identify the connecting maps in the defining direct system at the level of this module. By taking the colimit at the chain level, we obtain a general formula for colored knot Floer homology of an arbitrary knot in terms of its ordinary knot Floer homology and the colored unknot algebra. This extends the computation of Alishahi-Gorsky-Liu from L-space knots to all knots.

It is shown that the mapping class group of any non-sporadic surface admits a continuum of equivariant coarse median structures, thus improving a previous theorem of the author. The same holds for a wide class of right-angled Artin and Coxeter groups, including examples of any dimension at least two, and RACGs which are not quasi-isometric to RAAGs; the coarse median structures in these cases are not induced by cocompact cubulations. The results are instances of a general procedure, which uses quasicocycles to flexibly modify the structure of a (suitable) hierarchically hyperbolic group.

Let $S_g$ be the closed oriented surface of genus $g \geq 2$, and let $\Gamma$ be a finite collection of closed curves on $S_g$ that fills, in the sense that its complement is a union of disks. The total number of double points of such a $\Gamma$ in minimal position is at least $2g-1$; we call $\Gamma$ minimally intersecting when equality holds. We prove that the number $N_g$ of mapping class group orbits of minimally intersecting filling curve systems satisfies $N_g \sim 16^g (2g)!/(64\pi\sqrt{2}\, g^3)$ as $g \to \infty$. We also study the single-curve subproblem, in which $\Gamma$ consists of a single component. Every such curve is in particular a filling system, so the corresponding count $N_g^{(1)}$ satisfies $N_g^{(1)} \leq N_g$, which improves the upper bound previously known for single filling curves. Finally, we compute $N_g$ and $N_g^{(1)}$ exactly for small genus and conjecture an asymptotic growth rate.