math.GT daily digest: 17 submissions for 15 October 2026

New submissions

Ending currents of hyperbolic group actions

• Authors: Yair Minsky, Tianqi Wang
• arXiv: 2610.18887
• Subjects: Geometric Topology (math.GT)

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 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.

Exotic 's as unions of two thickened circles

• Authors: Sean Eli
• arXiv: 2610.19263
• Comments: 10 pages, 7 figures, No AI. Comments welcome!
• Subjects: Geometric Topology (math.GT)

We give an infinite family of non-diffeomorphic exotic '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 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 's using end Floer homology.

Two non-semisimple quantum invariants of lens spaces

• Authors: Rhea Palak Bakshi, Mitchell Jubeir, Quinn T. Kolt
• arXiv: 2610.19355
• Comments: 29 pages, 13 figures, comments welcome
• Subjects: Geometric Topology (math.GT); Category Theory (math.CT); Quantum Algebra (math.QA)

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

Relative knot concordance

• Authors: Vyacheslav Krushkal, Ryan Stees
• arXiv: 2610.19467
• Comments: 14 pages
• Subjects: Geometric Topology (math.GT)

Motivated by the relative-slice problem in $4$-manifold topology, we define for a knot $R$ the notion of a relative knot concordance group and analyze the structure of its algebraic analogue modulo torsion. Using Milnor signatures, we show that 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.

The orientation-reversing minimal stretch factor problem in small genus

• Authors: Pierre Dehornoy, Erwan Lanneau, Livio Liechti, Quentin Perroud
• arXiv: 2610.19828
• Comments: 25 pages, 1 figure, 12 tables
• Subjects: Geometric Topology (math.GT)

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.

On local minimality of Hausdorff dimension for round limit sets

• Authors: Sami Douba, Andrés Sambarino, Franco Vargas Pallete
• arXiv: 2610.20358
• Comments: 11 pages, 2 figures. Comments are welcome!
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

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.

Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds

• Authors: Oscar Ocampo
• arXiv: 2610.20478
• Comments: 27 pages. Comments are welcome
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

For , let be the crystallographic quotient of the Artin braid group, with translation subgroup , where . 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 . More precisely, for every , where is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup of dimension $N$ and a complete normal subgroup 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.

Minimal Complexes of Rational Tangles via the Fukaya Category

• Authors: Jonathan A. Higgins
• arXiv: 2610.20558
• Comments: 50 pages, many figures, 5 tables. Comments welcome!
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)

We compute minimal complexes for oriented rational tangles, denoted , 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.

Cross submissions

Motion groups, Legendrian links, and non-trivial loops of embeddings

• Authors: Gabriel Corrigan, Isacco Nonino
• arXiv: 2610.10804
• Cross-list: math.SG
• Comments: 20 pages, 0 figures. Comments welcome!
• Subjects: Symplectic Geometry (math.SG); Group Theory (math.GR); Geometric Topology (math.GT)

We introduce, for an embedded Legendrian link~$L$ in a contact 3-manifold~$(M,\xi )$, the group of its \emph{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.

Index for a mutation loop

• Authors: Tsukasa Ishibashi, Yuji Terashima
• arXiv: 2610.18940
• Cross-list: math.RT
• Comments: 31 pages
• Subjects: Representation Theory (math.RT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT)

Motivated by the 3d-3d correspondence, we introduce an index for a mutation loop $\varphi$ of a quiver and an integer . We prove its integrality 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 -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.

Enumeration of one-cycle signed chord diagrams and their duals for optimal Morse flows on closed surfaces

• Authors: Alexandr Prishlyak
• arXiv: 2610.19194
• Cross-list: math.CO
• Comments: 20 pages, 13 figures
• Subjects: Combinatorics (math.CO); Dynamical Systems (math.DS); Geometric Topology (math.GT)

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.

Algebraic overtwistedness of contact $(+1)$-surgeries via augmented Floer theories

• Authors: Russell Avdek, Youlin Li, Zhengyi Zhou
• arXiv: 2610.19333
• Cross-list: math.SG
• Comments: 36 pages, comments welcome!
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)

We prove that contact $(+1)$-surgery along a Legendrian $n$-sphere 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 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.

The Weil-Petersson diameter of moduli space for large genus

• Authors: Yunhui Wu
• arXiv: 2610.19707
• Cross-list: math.DG
• Comments: 27 pages. Comments welcome
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We determine the large-genus order of the Weil-Petersson diameter of for each fixed number of punctures: namely, . 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.

Drift and joint minimal displacement for groups acting on hyperbolic spaces

• Authors: Koji Fujiwara, Ryokichi Tanaka
• arXiv: 2610.19787
• Cross-list: math.PR
• Subjects: Probability (math.PR); Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)

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.

Mapping class groups of solenoidal manifolds

• Authors: Steven Hurder, Olga Lukina
• arXiv: 2610.20131
• Cross-list: math.DS
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

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.

Topological obstruction for Anosov representations into

• Authors: Kewei Li
• arXiv: 2610.20197
• Cross-list: math.DG
• Comments: 14 pages, comments welcome
• Subjects: Differential Geometry (math.DG); Group Theory (math.GR); Geometric Topology (math.GT)

We establish new restrictions on the virtual cohomological dimension of word hyperbolic groups admitting Anosov representations into () or the exceptional Lie group . Our main result states that, if is the parabolic subgroup of stabilizing an isotropic $k$-plane in , a group admitting a -Anosov representation into has virtual cohomological dimension strictly less than $p$. As a result, a -convex cocompact subgroup of cannot act properly discontinuously and cocompactly on . The main theorem also implies that a group admitting a -Anosov representation into the exceptional Lie group has virtual cohomological dimension strictly less than $4$, where $\alpha$ is a short restricted root of . As an application, we prove that the homogeneous space does not admit any compact quotient.

Completely parabolic surfaces are Veech

• Authors: Inbo Gottlieb Fenves
• arXiv: 2610.20566
• Cross-list: math.DS
• Comments: 2 pages + references
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

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