math.GT daily digest: 23 submissions for 3 November 2026

New submissions

On the second smallest ideal right-angled hyperbolic four-polytope

• Authors: Alexander Kolpakov
• arXiv: 2611.00118
• Comments: 18 pages; GitHub repository at this https URL
• Subjects: Geometric Topology (math.GT); Metric Geometry (math.MG)

The regular ideal right-angled 24-cell has volume and is the unique minimum-volume entirely ideal right-angled hyperbolic four-polytope. We prove that the next distinct volume is and that its unique equality case is the reflection double of the 24-cell across one octahedral facet. We also show that an entirely ideal right-angled four-polytope whose facets are regular ideal octahedra or their doubles across triangular faces decomposes into a chain of regular 24-cells, with opposite cutting facets in each intermediate cell.

Erratum to "Linear representations of 3-manifold groups over rings"

• Authors: Montek Singh Gill
• arXiv: 2611.00142
• Comments: 7 pages, including code; the final result remains exactly the same as in the original paper, only the argument changes. Erratum to arXiv:1703.06609
• Subjects: Geometric Topology (math.GT)

The fundamental groups of compact $3$-manifolds are known to be residually finite. Feng Luo conjectured that a stronger statement is true, by allowing only finite groups of the form where $R$ is a finite commutative ring with identity. In a prior paper, "Linear representations of $3$-manifold groups over rings", we, together with two co-authors, gave an equivalent formulation of Luo's conjecture via faithful representations, investigated variants of the conjecture where the target group is of type , or as in the original conjecture, and claimed to provide a $3$-manifold group which violated all four variants of the conjecture. We have discovered a flaw in the work, such that the existing proof is valid only for and . Here, we provide a modified argument for the and cases.

A nonarithmetic lattice from a mirror in the Couwenberg-Heckman-Looijenga construction

• Authors: Chenglong Yu
• arXiv: 2611.00270
• Comments: 17 pages, comments welcome!
• Subjects: Geometric Topology (math.GT)

We prove that the stabilizer of a reflection mirror of the Couwenberg-Heckman-Looijenga lattice induces a nonarithmetic lattice in $\mathrm{PU}(2,1)$. This lattice is not commensurable with any Deligne-Mostow lattice in $\mathrm{PU}(2,1)$ or any lattice in the twenty-two nonarithmetic classes listed by Deraux, Parker and Paupert.

Quantitative spectral rigidity for negatively curved manifolds and Outer Space

• Authors: Stephen Cantrell
• arXiv: 2611.00285
• Comments: 19 pages, 2 figures, to appear in Journal of Topology
• Subjects: Geometric Topology (math.GT)

Let $M$ be a closed manifold with fundamental group $G$. A subset of conjugacy classes is said to be spectrally rigid if when two negatively curved Riemannian metrics have the same marked length spectra on $E$, then their length spectra agree on all conjugacy classes. In this work we study spectrally rigid subsets from a quantitative perspective. We show that, if the density of , measured with respect to a fixed reference metric, decays more slowly than then $E$ is spectrally rigid. We establish analogous rigidity results for Culler-Vogtmann Outer Space and show that the threshold is sharp in this setting.

The recognition problem for 4-manifolds and the unknotting problem for 2-spheres in 4-manifolds

• Authors: Cameron Gordon
• arXiv: 2611.00355
• Comments: 15 pages
• Subjects: Geometric Topology (math.GT)

We show that there is no algorithm to decide whether or not a given 4-manifold is PL homeomorphic to the connected sum of 7 copies of S 2 x S 2 . We also show that the TOP and PL unknotting problems for all surfaces in the connected sum of 7 copies of S 2 x S 2 are unsolvable.

Minimal quasisections of circle bundles

• Authors: Ivan Kiselev, Gaiane Panina, Timur Shamazov
• arXiv: 2611.00679
• Subjects: Geometric Topology (math.GT)

Let be an oriented circle bundle over an oriented closed surface, and let $\mathcal{E}$ denote its Euler number. A \textit{quasisection} is a smooth surface $Q$, possibly with boundary, together with a generic smooth map such that . The \textit{local formula} expresses the Euler number $\mathcal{E}$ of a bundle as a sum of the weights of the \textit{singular vertices} of a quasisection. In the present paper determine the minimal number $\mathcal{N}$ of singular vertices for closed and for bordered quasisections for a given circle bundle. For bordered quasisections, [ \mathcal N_{\mathrm b}(E\to B) =\left\lceil\frac{3|\mathcal E|}{2}\right\rceil. ] For closed quasisections, If $\mathcal{E}$ is even, [ \mathcal N_{\mathrm cl}(E\to B) =\frac{3|\mathcal E|}{2}. ] If $\mathcal{E}$ is odd, [ \mathcal N_{\mathrm cl}(E\to B) =4|\mathcal E|. ] We also provide examples of quasisections that realize these minimal values.

On the nonvanishing of Floer lasagna modules

• Authors: Kai Zhu
• arXiv: 2611.00830
• Comments: 27 pages, 5 figures; comments welcome
• Subjects: Geometric Topology (math.GT)

In this paper, we establish several nonvanishing results for Floer lasagna modules for a smooth compact connected oriented $4$-manifold $W$ and a knot , via the evaluation maps to link Floer homology. We first establish a general criterion: if a certain Heegaard Floer cobordism map associated with a smooth $4$-manifold $W$ is nonzero, then , where is a local unknot. Applications include the nonvanishing of Floer lasagna modules for and for knot traces when $K$ is slice and $n$ is a non-positive integer. In another direction, we show that for every nontrivial knot $K$ and every integer $n$, the Floer lasagna module is nonzero, where is the dual knot. Moreover, if $K$ is an L-space knot, then is infinite-dimensional for all integers .

Casson invariant of knots defined by random braids

• Authors: Marina Ville
• arXiv: 2611.01188
• Subjects: Geometric Topology (math.GT)

We investigate the expectation of the Casson invariant for knots represented by two similar types of braids, the rosette $s$-braids which are products of $k$ factors and the quasitoric $s$-braids which are products of $k$ factors . If $k$ is of the form , where $c$ is the number of crossing points of braid; if $k$ is of the form , .\ If we go through all $3$-braids, regardless of the shape of their diagram, we get . \ For knots with a checkerboard diagram, .

Lehmer's number is not an orientable stretch factor in genus six

• Authors: Lizi Guo
• arXiv: 2611.01350
• Comments: 22 pages, 10 figures. Code at this https URL
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

Let be Lehmer's number. We prove that no pseudo-Anosov homeomorphism of a closed orientable surface of genus six with orientable invariant foliations has stretch factor . This gives a negative answer to a question of Lanneau and Thiffeault on the minimal stretch factor in even genus. As a consequence, the minimal stretch factor of such maps is the largest root of , approximately $1.20189$. The proof combines the Lefschetz method of Lanneau and Thiffeault, the standardly embedded train tracks of Hironaka and Tsang, an elementary analysis of the transition digraph, and an obstruction from periodic conjugacy classes in the fundamental group.

Dimension bounds for finite-volume hyperbolic Coxeter polytopes

• Authors: Lizi Guo, Fangting Zheng
• arXiv: 2611.01795
• Comments: 77 pages
• Subjects: Geometric Topology (math.GT); Combinatorics (math.CO); Metric Geometry (math.MG)

We prove that a finite-volume hyperbolic Coxeter polytope has dimension at most $81$, substantially improving Prokhorov's bound of $995$. A refinement of Prokhorov's method gives the intermediate bound $162$, while an extension of Vinberg's planar-angle comparison to the finite-volume setting yields the sharper bound. Both arguments combine improved weighting functions with estimates using cusp geometry to control weight contributions of angles at ideal vertices or edges incident to them.

Braided Multisections and Aspherical Stein Fillings

• Authors: Anar Akhmedov
• arXiv: 2611.01833
• Comments: 24 pages, 3 figures
• Subjects: Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We construct Stein fillings diffeomorphic to complements of a connected symplectic unbranched multisection and a regular sphere fiber in the ruled surface . For every and every multisection degree , a fixed contact three-manifold admits infinitely many pairwise nonhomeomorphic aspherical fillings of this form. Their first and second integral homology groups are and , respectively, and their intersection pairings vanish. We vary the braid monodromy while preserving the marked contact boundary. Unbounded torsion orders in the first homology of regular unbranched covers of degree detect infinitely many isomorphism types of fundamental groups. We also give an equivariant Weinstein handle construction for the cyclic filling . The contact boundaries are supported by spinal open books with planar pages and have positive ordinary support genus.

Relative trisection maps on 4-manifolds

• Authors: Osamu Saeki
• arXiv: 2611.02494
• Comments: 19 pages, 13 figures
• Subjects: Geometric Topology (math.GT)

We show that for an arbitrary compact orientable smooth $4$--dimensional manifold $M$ with a given open book structure on the boundary , which may be disconnected, there exists a generic smooth map of $M$ onto the $2$--disk which gives a relative trisection on $M$ consistent with the given open book structure on $\partial M$. This gives a new proof to the existence of a relative trisection on $M$. In fact, it turns out that every relative trisection can be obtained in this way. As a byproduct, we also show that $M$ always admits a decomposition into a PALF (positive allowable Lefschetz fibration) over the $2$--disk and a fiber bundle over the circle with a sutured compression body fiber, consistent with the given open book structure on $\partial M$ and with the surface bundle structure of the PALF over .

Quasimorphisms and bounded cohomology of transformation groups of nonorientable surfaces

• Authors: Michael Brandenbursky
• arXiv: 2611.02847
• Comments: 36 pages, one figure
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Symplectic Geometry (math.SG)

Let be a closed connected nonorientable surface of genus , let $\omega$ be a smooth positive density, and let $\mu$ be a good measure. In this paper we generalize the Gambaudo--Ghys construction using homogeneous quasimorphisms on the central quotients of the braid groups of . We prove injectivity of this map for density-preserving diffeomorphisms. For and quasimorphisms vanishing on the disc braid group, we obtain an injective map to -continuous homogeneous quasimorphisms on . We also obtain undistortion criteria in and . For and every , we prove $$ \dim\overline H_b^d(Homeo_0(N_g,\mu))=\infty,\qquad \dim\overline{EH}_b^d(Diff_0(N_g,\omega))=\infty. $$ Moreover, is infinite dimensional in every positive even degree $d$. Here and denote reduced bounded cohomology and reduced exact bounded cohomology, respectively.

Multivariable Alexander polynomials of twisted torus links via Dehn surgery

• Authors: Birch Bryant
• arXiv: 2611.03265
• Subjects: Geometric Topology (math.GT)

Let be the twisted torus link obtained from the torus link by adding $s$ full twists to $r$ adjacent strands, and let $C$ be an unknotted circle encircling those strands. For all $p$, $q$, and $r$ we compute the multivariable Alexander polynomial of in closed form. A relation of Silver and Williams between the Alexander polynomials of a link and of its $1/n$ Dehn fillings then gives a closed formula for the multivariable Alexander polynomial of every twisted torus link , answering a question of Adnan and Park. When , so that is a knot, the formula is a sum of $pr$ monomials with coefficient $+1$, divided by cyclotomic factors, and its exponents are governed by an elementary counting function on the orbit of $q$ modulo $p$. Because the numerator has no cancellation, we are able to give the degree for all $s$ and a simple non-fiberedness criterion.

Cross submissions

Blowdowns along Ricci flow in dimension four

• Authors: Dain Kim, Tristan Ozuch
• arXiv: 2611.00139
• Cross-list: math.DG
• Comments: 48 pages
• Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Geometric Topology (math.GT)

We show that the blowdown shrinking Ricci soliton of Feldman, Ilmanen and Knopf is strictly dynamically stable in the following strong local sense:

(1) The Ricci flow from every metric on a closed 4-manifold containing a region close to a rescaling of a sufficiently large compact subset of the soliton in either orientation and with controlled geometry elsewhere develops a singularity with unique tangent flow modeled on the soliton.

(2) The singular time slice has an isolated conical singularity, and by Gianniotis-Schulze the flow can be continued by blowing down a sphere of self-intersection $\pm 1$.

(3) On a closed 4-manifold, the set of metrics whose Ricci flows have their first singularity modeled on the soliton is -open. It is nonempty if and only if the manifold contains a sphere of self-intersection $\pm 1$.

(4) The soliton is strongly rigid among shrinking Ricci solitons.

As applications, we construct -open sets of initial data whose Ricci flows perform any prescribed ordered sequence of blowdowns.

(1) Every closed simply connected smooth 4-manifold, including spin and exotic examples, can be reached by Ricci flow from a standard , through finitely many blowdown singularities.

(2) On the standard , there are infinitely many open sets of initial metrics whose Ricci flows, after a single blowdown, reach pairwise nondiffeomorphic manifolds.

The proof uses the spectral analysis of Naff and the second author and deals with the gauge-fixing problem along Ricci flow through a Ważewski argument inspired by Stolarski. We also use Bamler-Lai's PDE-ODI principle. Except for continuation (requiring nonnegative curvature of the asymptotic cone), all results apply to linearly stable nonflat asymptotically conical shrinkers in all dimensions.

There is only one transitive Anosov flow (in dimension 3, with orientable foliations, up to bicontact padding and Goodman surgeries)

• Authors: Thomas Barthelmé, Federico Salmoiraghi, Chi Cheuk Tsang
• arXiv: 2611.00977
• Cross-list: math.DS
• Comments: 70 pages, 18 figures
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT); Symplectic Geometry (math.SG)

We introduce a new surgical operation on Anosov flows, which we call bicontact padding surgery. The operation is defined using the machinery of strongly adapted bicontact structures, and involves inserting a 'padding' layer where one of the contact structures rotates many times around the Reeb flow of the other. We show that every skew Anosov flow with orientable stable and unstable foliations can be transformed into a finite cover of a geodesic flow using bicontact padding surgery and Goodman surgery. Together with previous results, this implies that every transitive Anosov flow with orientable stable and unstable foliations can be transformed into a suspension flow using bicontact padding surgery and Goodman surgery. This gives some progress towards the Christy-Fried-Ghys conjecture, which predicts that the same statement is true using only Goodman surgery.

Generating the liftable outer automorphism groups of regular cyclic covers

• Authors: Rohan Suresh Mahure, Priyavrat Pandey, Kashyap Rajeevsarathy
• arXiv: 2611.01444
• Cross-list: math.GR
• Comments: 19 pages, 2 figures
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

Let be a finite connected one-dimensional CW complex with , and let be the outer automorphism group of the free group , where . Let be the standard $k$-sheeted regular cyclic cover of induced by a -action on (which is homotopy equivalent to , where ) generated by a free rotation through $2\pi /k$ of . Let be the liftable outer automorphism group associated with a standard $k$-sheeted normal cover . For , we show that forms an infinite family of self-normalizing subgroups of . For , these groups are maximal if and only if $k$ is prime. Furthermore, we derive explicit finite generating sets for for and . Let $\mathcal{C}$ be the infinite chain of roses, and let be the standard infinite-sheeted cover induced by $⟨\tau ⟩$, where $\tau$ is the unit translation on $\mathcal{C}$. As a final application, we obtain a finite generating set for when .

A faithful representation and profinite completion

• Authors: Wonyong Jang
• arXiv: 2611.01672
• Cross-list: math.GR
• Comments: 16 pages. Comments are welcome!
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Representation Theory (math.RT)

We exhibit two finitely generated residually finite groups and with isomorphic profinite completions , such that has a faithful representation into but is not linear. This answers a question of Andrei Jaikin-Zapirain, Alexander Lubotzky, and Martin Bridson negatively. We also present two finitely generated residually finite groups and with isomorphic profinite completions satisfying the following. For every field $k$ with characteristic and transcendence degree , there exists such that has a faithful representation into . However, admits no faithful finite-dimensional representation over any field of positive characteristic.

Partially wrapped contact Fukaya categories of 1-jet spaces

• Authors: Hanming Liu
• arXiv: 2611.01689
• Cross-list: math.SG
• Comments: 42 pages, 12 figures
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We introduce the partially wrapped contact Fukaya category of a 1-jet space equipped with a stop, which is an -category of Legendrians equipped with representations of their Chekanov-Eliashberg algebras. This category is invariant under contact isotopies of the stop. We prove a wrapping exact triangle and a generation theorem. When the stop is bounded subanalytic singular isotropic, we prove a Morita equivalence between the partially wrapped contact Fukaya category and a microlocal sheaf category.

Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

• Authors: Haozhao Li, Bing Wang, Kai Zheng
• arXiv: 2611.01882
• Cross-list: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We study global Calabi flows $g(t)$ on compact Kähler surfaces with a uniform Sobolev constant. For any sequence , set . After passing to a subsequence, these flows converge to an eternal Calabi flow on a compact extremal Kähler orbifold surface. The convergence holds in the spacetime -Cheeger--Gromov topology: it is uniform in the Gromov--Hausdorff sense on compact time intervals and smooth away from the orbifold singularities, using comparisons fixed in time.

Boundary fibers of the holonomy map for complex projective structures

• Authors: Shiyu Cao, Sheng Tan
• arXiv: 2611.01928
• Cross-list: math.DG
• Comments: 34 pages
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We study the Morgan--Shalen boundary of holonomy representations of complex projective structures on a compact Riemann surface of genus at least two. We prove that every direction defined by a holomorphic quadratic differential with a multiple zero admits distinct projective marked length limits. Together with Dumas's uniqueness theorem for simple zeros, this gives a complete characterization of directions with unique limits. We also establish that no nonzero abelian length function can occur in the holonomy boundary of any such fixed surface, regardless of the limiting quadratic differential. Finally, we characterize all directional boundary fibers by a necessary-and-sufficient analytic condition involving valued representations and the holomorphic jet bundle underlying an oper.

Complete Topological Invariant and Coding of Homoclinic Flows on the Two-Dimensional Sphere

• Authors: Alexandr Prishlyak, Illia Ovtsynov
• arXiv: 2611.02116
• Cross-list: math.DS
• Comments: 19 pages, 5 figures
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

We study codimension one homoclinic flows on closed surfaces and introduce a complete topological invariant based on the structural graph of the flow and the distinguishing graphs of its atoms. This invariant completely determines the topological equivalence class of a homoclinic flow. For flows on the two-dimensional sphere, we construct a canonical coding algorithm that avoids the global renumbering required in the classical coding of Morse flows and substantially reduces the computational complexity. As an application, we derive recursive formulas for the number of topologically distinct homoclinic flows on the two-dimensional sphere and compute these numbers for flows with a small number of saddle singularities and closed orbits.

-rigidity of -convex cocompact representations of Gromov hyperbolic groups

• Authors: Nikolay Bogachev
• arXiv: 2611.03539
• Cross-list: math.DG
• Comments: 33 pages, comments are very welcome!
• Subjects: Differential Geometry (math.DG); Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)

Let and let $\Gamma$ be a non-elementary Gromov hyperbolic group with . We prove that every -convex cocompact representation $$ \rho:\Gamma\longrightarrow \mathrm{PO}_{p,3} $$ whose proximal limit set is a submanifold preserves a totally geodesic copy of . Consequently, $\Gamma /\ker \rho$ is virtually a cocompact lattice in .

For odd $p$, we prove the same conclusion for every signature $(p,q)$ with , under the same hypothesis .

Our methods combine dynamics, differential topology, and representation theory. The key new ingredient is to study the radicals of the semidefinite forms induced on the tangent spaces of the limit set. They form a continuous vector bundle of constant rank. The tangent action at attracting fixed points gives a bound, in terms of the radical rank, on the number of eigenvalues of an orthogonal lift whose modulus exceeds one. We then use the topology of the sphere and the representation theory of real algebraic groups to exclude the remaining possibilities. In arbitrary signature with , these arguments already imply that a representation with a limit set cannot be Zariski dense.