Papers

We prove that completely parabolic translation surfaces are Veech, answering a question of Smillie-Weiss.

We compute minimal $\mathfrak{sl}_N$ complexes for oriented rational tangles, denoted $[\![\tau_{u/v}^{O}]\!]_N$, in terms of the topological Fukaya category of the $3$-punctured disk. This demonstrates that the complexes stabilize, allowing us to define a HOMFLY-PT homology for rational tangles.

For $n\geq3$, let $\overline B_n=B_n/[P_n,P_n]$ be the crystallographic quotient of the Artin braid group, with translation subgroup $\overline P_n=P_n/[P_n,P_n]\cong\mathbb Z^N$, where $N=\binom n2$. We prove that every compact connected flat $3$-manifold $M$, considered up to affine equivalence, is the base of a geometric torus bundle whose total space is a compact flat manifold with fundamental group a Bieberbach subgroup of some $\overline B_n$. More precisely, for every $n\geq n_M$, where $n_M\in\{3,4,5\}$ is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup $\Gamma_n(M)\leq\overline B_n$ of dimension $N$ and a complete normal subgroup $K_n(M)\cong\mathbb Z^{N-3}$ consisting of translations such that $$ \Gamma_n(M)/K_n(M)\cong\pi_1(M). $$ We also determine which of the associated torus-bundle group extensions split: for the realizations constructed here, splitting occurs for exactly four of the ten affine-equivalence classes of compact flat $3$-manifolds.

We address a question of Canary–Taylor and Canary–Minsky–Taylor for sufficiently symmetric compact hyperbolic $3$-orbifolds with totally geodesic boundary. We give sufficient conditions on the symmetries of such an orbifold to show that, among the complete convex cocompact structures on the interior, the structure with Fuchsian ends is a critical point (respectively, a strict local minimum) for the critical exponent. We also provide an infinite family of non-rigid examples fulfilling these conditions.

We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into $\operatorname{PO}(p, q)$ ($p > q \geq 2$) or the exceptional Lie group $\operatorname{G}_2'$. Our main result states that, if $P_k$ is the parabolic subgroup of $\operatorname{PO}(p, q)$ stabilizing an isotropic $k$-plane in $\mathbb{R}^{p, q}$, a group admitting a $\{P_1,P_q\}$-Anosov representation into $\operatorname{PO}(p, q)$ has virtual cohomological dimension strictly less than $p$. As a result, a $\mathbf{H}^{p, q-1}$-convex cocompact subgroup of $\operatorname{PO}(p, q)$ cannot act properly discontinuously and cocompactly on $\mathbf{H}^{p, q-1}$. The main theorem also implies that a group admitting a $P_{\alpha}$-Anosov representation into the exceptional Lie group $\operatorname{G}_2'$ has virtual cohomological dimension strictly less than $4$, where $\alpha$ is a short restricted root of $\operatorname{G}_2'$. As an application, we prove that the homogeneous space $\operatorname{G}_2'/ \operatorname{SU}(2, 1)$ does not admit any compact quotient.

The goal of this work is to calculate the mapping class groups of solenoidal manifolds. We introduce the dynamical commensurator group associated to an odometer action, and develop methods to calculate the mapping class group using the dynamical commensurator. Our results generalize the previous results of Odden for the mapping class group of the universal solenoid over a Riemann surface, and of Bering and Studenmund for the pointed mapping class group of the universal solenoid over a compact aspherical manifold. We include a selection of examples.

We determine the minimal stretch factors among all orientation-reversing pseudo-Anosov mapping classes of the closed orientable surfaces of genus two, three and four.

We obtain a uniform lower bound for the drift of random walks on hyperbolic spaces in terms of joint minimal displacement. Our main theorem applies to weakly acylindrical actions on Gromov-hyperbolic spaces and is quantitative in all relevant constants. For discrete isometry groups of Hadamard manifolds with pinched negative curvature, the required assumptions are automatically satisfied by the Margulis lemma. This yields uniform gaps at zero in drift spectra for pinched negatively curved Hadamard manifolds, hyperbolic groups, and mapping class groups acting on curve graphs.

We determine the large-genus order of the Weil-Petersson diameter of $\mathcal{M}_{g,n}$ for each fixed number of punctures: namely, $\mathrm{diam}(\mathcal{M}_{g,n})\asymp\sqrt g\log g$. The main new ingredient is an integrated Ricci estimate for geodesics that minimize in moduli space and have thick endpoints, with no systole bound along their interiors.

Motivated by the relative-slice problem in $4$-manifold topology, we define for a knot $R$ the notion of a relative knot concordance group $\mathcal{C}_R$ and analyze the structure of its algebraic analogue $\mathcal{AC}_R$ modulo torsion. Using Milnor signatures, we show that $\mathcal{AC}_R$ has infinite rank for every $R$. We also show that any prescribed finite set of natural numbers can be realized as orders of elements in a single relative concordance group.

We calculate the chromatic spherical invariant $\mathscr{K}_{\mathrm{Rep}(H)}(L(p,q))$ of lens spaces $L(p, q)$ for arbitrary finite-dimensional unibalanced, unimodular Hopf algebras $H$. In the case of the index-2 Nichols Hopf algebra $\mathcal{K}_2$ and $q=1$, we find that the invariant of $L(p, 1)$ depends only on the parity of $p$ for $p\geq 1$. We also describe the duals $\mathcal{S_I}(M)^*$ of the admissible skein modules of handlebodies $M=H_g$ and of $M=L(p, q)$ for arbitrary tensor ideals $\mathcal{I}$ of a ribbon tensor category. In particular, $\mathcal{S_I}(L(p, q))^*$ corresponds with cyclic traces on $\mathcal{I}$ satisfying a property reminiscent of the modified trace condition.

We prove that contact $(+1)$-surgery along a Legendrian $n$-sphere $\Lambda\subset(\mathbb{S}^{2n+1},\xi_{\mathrm{std}})$ with \[ \mathrm{tb}(\Lambda)\ne (-1)^{\frac{n(n-1)}{2}+1} \] has vanishing contact homology. In dimension three, this produces new examples of tight algebraically overtwisted contact manifolds, vastly extending the authors' previous results: If a Legendrian knot in $(\mathbb{S}^3,\xi_{\mathrm{std}})$ admits an exact, oriented Lagrangian filling that is not a disk, then contact $(+1/k)$-surgery on it is algebraically overtwisted and tight for every positive integer $k$. Our proofs use versions of symplectic cohomology and wrapped Floer cohomology defined by augmentations of a chain-level contact homology algebra.

We give an infinite family of non-diffeomorphic exotic $\mathbb{R}^4$'s, each of which is the union of two copies of a standard open tubular neighborhood of the circle, which intersect in a standard open tubular neighborhood of the wedge of two circles. This gives the simplest known decomposition of an exotic $\mathbb{R}^4$ into two open sets, making progress on Question 4.43 (A) of Kirby's 1995 problem list. The decompositions are obtained by studying fibered slice disk complements in certain Casson-Gordon homotopy 4-balls, for which we provide similar decompositions. We distinguish the $\mathbb{R}^4$'s using end Floer homology.

Optimal Morse vector fields on closed surfaces, i.e.\ fields with the minimal possible number of critical points (a single source and a single sink), are classified up to topological trajectory equivalence by one-cycle signed chord diagrams in Kybalko, Prishlyak and Shchurko (2018). We recall this construction and the associated duality, which corresponds to reversing the direction of the flow, in a unified encoding in which both endpoints of a chord carry the same label, and we make it algorithmic. We give an explicit traversal procedure that detects one-cycle diagrams, a canonical form of an equivalence class under cyclic shifts, reversal and relabeling of chords, and a construction of the dual diagram. On this basis we describe an algorithm that enumerates all one-cycle signed chord diagrams with a given number of chords, both for orientable and for non-orientable surfaces, groups them into equivalence classes, pairs each class with its dual, and identifies the self-dual ones. The algorithm is implemented in Python and archived in Zenodo; its output is validated against the Harer–Zagier numbers in the orientable case and against orbit-size counts in both cases. We also note that, when a one-cycle diagram is read as a cellular embedding of a one-vertex graph, its dual diagram describes the embedding of the dual graph.

Motivated by the 3d-3d correspondence, we introduce an index $\mathcal{I}_g^\phi$ for a mutation loop $\phi$ of a quiver and an integer $g \geq 0$. We prove its integrality $\mathcal{I}_g^\phi \in \mathbb{Z}$ under certain conditions, which extends the integrality conjecture of Gang–Kim–Yoon for $3$-manifolds to the broader framework of mutation loops in cluster algebras. Our index also has an interpretation as the genus-$g$ partition function of a $(1+1)$-dimensional TQFT. We provide explicit computations for mutation loops of period one. In particular, in the case of the generalized Kronecker quivers, we give an explicit formula expressing the generating function of the indices for all genera $g$ as a ratio of polynomials. Altogether, we provide a new insight to the interplay between the geometry of $3$-manifolds and cluster algebra.

We prove that closed arithmetic hyperbolic $n$-manifolds of bounded rank have bounded adjoint trace field degree, and consequently have lower bounded injectivity radius. From this, we obtain a number of corollaries. In particular, we obtain that closed arithmetic hyperbolic $3$-manifolds of bounded rank are contained in finitely many commensurability classes, and describe these commensurability classes. Additionally, it follows that for closed arithmetic hyperbolic $3$-manifolds, Heegaard genus is equal to rank for all but finitely many manifolds of each rank.

Building on work of Kauffman–Saleur and earlier work of the second author, we derive a determinant formula for the Links–Gould polynomial of links. As an application, we prove a 20-year-old conjecture of Ishii: for any alternating link, the Links–Gould invariant is an alternating polynomial.

In this short note we show the existence of stably exotic knotted surfaces in 4-manifolds in the sense that the embedded surfaces are topologically equivalent but are not smoothly equivalent even after any number of internal and external stabilisations.

We prove the Labastida–Mariño–Ooguri–Vafa conjecture for oriented links with arbitrary integer framings. Consequently, the framed reformulated invariants have finite integral LMOV expansions for every integer framing vector.

The stability zone of Novikov's problem of planar sections of triply periodic surfaces are labeled by indivisible triples of integers, which we call the soul of the zone. Which triples may occur is not known. We show that, if a Morse function $f$ on $\mathbb T^3$ admits a translation $\tau(x)=x+v$ with $f\circ\tau=-f$, so that the sets $\{f<0\}$ and $\{f>0\}$ are congruent, then in the generic genus-$3$ situation every topological number $L$ satisfies $\langle L,2v\rangle\equiv1\pmod2$. The mechanism is that the electron/hole type of a null-homologous closed trajectory is invariant under the period lattice and is reversed by $\tau$. For $f=\cos2\pi x+\cos2\pi y+\cos2\pi z$ and the $\mu$-cube, this means that the sum of the components of each soul is odd.

We prove that the rational Chow ring of every connected component of the multi-scale compactifications of the genus three strata $\mathcal{H}(4), \mathcal{H}(2,2)$ and $\mathcal{H}(3,1)$ is generated by boundary-stratum classes. Our proof shows Chow-Künneth generation property of boundary strata of the multi-scale compactification by reducing the boundary analysis to lower-genus generalized strata of meromorphic differentials, including residueless loci, described through plane-quartic and plane-cubic models. As a consequence, the rational cycle-class map is an isomorphism and the compactifications have no odd rational cohomology.

We prove that the fundamental group of every compact graph $3$-manifold is linear over the complex numbers. As a corollary, all fundamental groups of compact $3$-manifolds are linear over the complex numbers.

A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If $G$ is an infinite countable group, and $X$ is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on $X$ admitting a gpnf-action by $G$. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus $g \geq 2$. If $G$ is a finite group, then its purely non-free genus $\sigma^{pnf}_{hyp}(G)$ is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by $G$. This parameter is already known for cyclic, abelian non-cyclic groups, ${\mathcal A}_{4}$, ${\mathcal A}_{5}$ and ${\mathfrak S}_{4}$. For general $G$, we provide the value of $\sigma^{pnf}_{hyp}(G)$, correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups $G^{0}_{s}(n)=\langle x,y: x^{n}=y^{2}=1, yxy=x^{s}\rangle,$ where $n \geq 3$ and $2 \leq s \leq n-1$ is such that $s^{2} \equiv 1 \mod(n)$.

We study the behavior of saddle connection directions on translation surfaces from the point process viewpoint. We define a direction point process for saddle connections of length at most $R>0$ on every translation surface. Given a general ergodic $\mathrm{SL}(2,\mathbb R)$-invariant measure $\mu$ on a stratum of translation surfaces, we also define a slope point process associated with $\mu$. By utilizing the dynamics and geometry of strata, we show that for almost every translation surface, the direction point processes, rescaled by $R^2$, converge in distribution to the slope point process as $R\to\infty$. Using Palm distribution theory, we prove the existence of limiting joint gap distributions and obtain new regularity results for gap distributions associated with the general $\mu$. Finally, we study how these point processes and joint gap distributions behave as the genus $g$ tends to infinity. In the principal stratum, we show that the slope point processes, rescaled by $g^2$, converge in distribution to a Poisson point process. By passing to the corresponding Palm distributions, we show that the rescaled slope gaps converge jointly to independent exponential random variables, which is in stark contrast to the known non-exponential gap distributions in a fixed genus.

A teardrop geodesic on a hyperbolic surface is a closed curve that is geodesic except at exactly one point, called its vertex. One imagines a closed curve with one point held down as it deforms to a geodesic. For each teardrop geodesic, we give an explicit geometric construction of the closed geodesic in its free homotopy class. This construction also yields a relation between the length of the teardrop geodesic, the angle at its vertex, and the length of its geodesic representative. We then apply the construction to a self-intersecting closed geodesic at a chosen self-intersection point. Since, at that point, the geodesic splits into two directed teardrop geodesics, our construction gives a geometric description of Turaev's cobracket in terms of geodesic representatives. Using this description we prove that if two self-intersection points yield a pair of canceling terms in the cobracket, then the angle at these points and the lengths of the small loops determined are the same.

Ideal Poisson–Voronoi tessellations are random tessellations that arise as limits of Voronoi diagrams for Poisson point processes as their intensity tends to zero. For the Riemannian metric on a symmetric space of higher rank, Frączyk, Mellick, and Wilkens (arXiv:2307.01194) proved that the corresponding ideal Poisson–Voronoi tessellation is almost surely completely connected, with an unbounded side between any two cells. This contrasts sharply with the geometry of ideal Poisson–Voronoi tessellations in hyperbolic spaces studied by D'Achille, Curien, Enriquez, Lyons, and Ünel (arXiv:2303.16831). In this paper we introduce and study ideal Poisson–Voronoi tessellations for Finsler metrics on higher rank symmetric spaces. We show that ideal Poisson–Voronoi tessellations with respect to polyhedral Finsler metrics can exhibit either behavior seen in these previous works: generically they resemble the hyperbolic case, where almost surely all cells have bounded sides. We further introduce the ideal Selberg–Poisson–Voronoi tessellation of the space of positive definite symmetric matrices, which has particularly nice properties.

Borzellino and Brunsden introduced a general notion of suborbifold and proved that a suborbifold is the image of a complete orbifold embedding if and only if it is both saturated and split. Their definitions and proofs are formulated in the classical language of orbifold charts. In this paper we translate these notions into the language of proper étale Lie groupoids.We introduce the groupoid analogues of saturation and splitting, which we call orbit saturation and stabilizer splitting, and prove that they coincide with the chart-based conditions. Our main theorem states that a subgroupoid is the image of a complete orbifold embedding if and only if it is both orbit saturated and stabilizer split. We then prove that these notions are Morita invariant: if two orbifold groupoids are Morita equivalent, then their orbit saturated and stabilizer split subgroupoids correspond naturally to each other. This solves an open problem raised by Mestre and Weilandt and explains why the embedding definition of Cho–Hong–Shin fails to be Morita invariant. Along the way we provide detailed examples, including the diagonal embedding, the non-split example of Borzellino and Brunsden, and explicit Morita equivalences.

We obtain very high genus asymptotics for double Hurwitz numbers when both the size $n$ and the genus $g$ go to infinity, under the constraint that $g$ grows much faster than $n$, more precisely that $\frac{2g}{n}-\ln n\to\infty$. Our proof relies on adapting arguments about the mixing time of the transposition random walk introduced by Diaconis and Shahshahani.

Let $f$ be a homeomorphism of a closed orientable surface of genus at least one that is isotopic to a pseudo-Anosov homeomorphism, possibly relative to a finite invariant set. We construct an intrinsic monotone factor by collapsing the maximal continua whose iterates have uniformly bounded diameter in the universal cover. The quotient is homeomorphic to the original surface and the factor is isotopic to $f$ and has several strongly chaotic dynamical features. It is infinitely continuum-wise expansive, tight, and topologically mixing, and it has a dense family of rotational horseshoes, dense periodic points, and a single topological homoclinic class. The factor can be taken as an area-preserving homeomorphism. The quotient preserves stable translation length on the fine curve graph and the applicable homological rotation sets. When $f$ is $C^\infty$, a companion paper gives a unique measure of maximal entropy for the factor; it has full support and is Bernoulli.

The canonical component of the character variety of a hyperbolic 2-bridge link complement M is an algebraic surface, from the work of Thurston. If we fix a boundary component T of M and perform a Dehn-filling on T, we get a 1-cusped manifold whose character variety is commonly a generalized curve that is naturally embedded in the character variety of M. If we perform distinct Dehn-fillings on M, we get various generalized curves on the canonical surface. In this paper, we analyze how these curves intersect with each other in the surface and study the topological implications for these intersection points. We also classify the intersection points that correspond to reducible characters, using the twisted Alexander polynomials associated to the Dehn-fillings. Throughout the paper, we use the Whitehead link as a motivating example to illustrate our core methodology, while some generalizations to other 2-bridge links will also be discussed.