math.GT daily digest: 11 submissions for 24 September 2026

The Thursday, 24 September 2026 arXiv math.GT new-submissions listing contains 11 eligible papers: 3 new submissions and 8 cross submissions. Replacement submissions are excluded.

New submissions

Geometric ideal triangulations of hyperbolic 3-manifolds

Authors: Huabin Ge
arXiv: 2609.27635
Comments: 20 pages, 3 figures. Comments are welcome
Subjects: Geometric Topology (math.GT)

We resolve the longstanding geometric ideal triangulation conjecture arising from Thurston's theory of hyperbolic 3-manifolds: every finite-volume cusped hyperbolic 3-manifold admits a finite geometric ideal triangulation realizing its complete hyperbolic structure.

Official arXiv abstract page

Counting Invariants for Torus Quotients under FPF Group Actions

Authors: Xue Zhang
arXiv: 2609.27736
Comments: 19 pages
Subjects: Geometric Topology (math.GT)

Let $G$ be a finite group admitting a rational fixed-point-free (FPF) representation $V$ , with a $G$ -invariant full-rank lattice . We study the induced $G$ -action on the complex torus and the quotient orbifold . We prove that every FPF group admits a unique irreducible rational FPF representation. We derive closed-form formulas for three core geometric invariants: the number of non-free points of the torus action, the number of singular points of the quotient orbifold, and its orbifold Euler characteristic.

Official arXiv abstract page

Finiteness of closed arithmetic hyperbolic surface bundles

Authors: Tam Cheetham-West; Homin Lee; Nicholas Miller
arXiv: 2609.28188
Comments: 23 pages, 3 figures. Comments welcome!
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

For each , we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$ -manifolds with genus $g$ fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL whose image is contained in the fundamental group of some arithmetic hyperbolic $3$ -manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.

Official arXiv abstract page

Cross submissions

Algebraically primitive Teichmüller curves in Ω\mathcal M_g(2g-2)^{\mathrm{hyp}}

Authors: Myeongjae Lee
arXiv: 2609.26903
Cross-list: math.DS
Comments: 17 pages, comments are welcome
Subjects: Dynamical Systems (math.DS); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We give a complete classification of algebraically primitive Teichmüller curves in for . They are precisely the curves generated by the double regular -gon when is prime, and by the regular $4 g$ -gon when $g$ is a power of two.

Official arXiv abstract page

Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces

Authors: Junehyuk Jung; Insung Park; Peter Zenz
arXiv: 2609.27071
Cross-list: math.DS
Comments: 66 pages, 5 figures
Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG); Geometric Topology (math.GT); Spectral Theory (math.SP)

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients uniformly in and , where and are the eigenparameters corresponding to Maass forms $ϕ$ and $φ$ on $X$ .

Official arXiv abstract page

Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

Authors: Krishnendu Gongopadhyay; Neelanjan Mondal
arXiv: 2609.27125
Cross-list: math.DS
Comments: Preliminary version
Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

Let be a Zariski-dense Borel-Anosov subgroup and let $φ$ be positive on its limit cone. For every , we construct twisted $(φ, χ)$ -conformal densities and prove their uniqueness, atomlessness, and ergodicity. For every normal subgroup with nilpotent quotient, we classify the ergodic $φ$ -conformal densities of in terms of characters of .

Official arXiv abstract page

Finite quotients of spherical Artin groups

Authors: Sam Hughes; Thomas Ng; Kaitlin Ragosta; Nancy Scherich; Yvon Verberne
arXiv: 2609.27171
Cross-list: math.GR
Comments: This is a preliminary preprint which we intend to update in the near future
Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We show the smallest non-abelian quotients of spherical and affine Artin groups are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. We deduce irreducible spherical Artin groups are determined by their finite quotient groups.

Official arXiv abstract page

Bounded cohomology, Codimension two submanifolds and Pontryagin-Thom constructions

Authors: Thorben Kastenholz
arXiv: 2609.27476
Cross-list: math.AT
Comments: This paper represents a rework of the erroneous paper 2503.22511. It represents the attempts of the author to repair and further develop the approach of 2503.22511
Subjects: Algebraic Topology (math.AT); Group Theory (math.GR); Geometric Topology (math.GT)

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

Official arXiv abstract page

Modular integrals for individual modular knots

Authors: Toshiki Matsusaka
arXiv: 2609.27574
Cross-list: math.NT
Comments: 12 pages
Subjects: Number Theory (math.NT); Geometric Topology (math.GT)

To each modular knot, we attach a weight $2$ modular integral for whose homogenized cycle integrals recover its linking numbers with other modular knots in . This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$ . Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoglu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.

Official arXiv abstract page

Milnor's inequality and circular elastic knots

Authors: Tatsuya Miura
arXiv: 2609.27643
Cross-list: math.DG
Comments: 14 pages
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Geometric Topology (math.GT)

We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the -closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter--von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti--Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach--Reiter--von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.

Official arXiv abstract page

Geometries of Quantum Field Theories

Authors: Masahito Yamazaki
arXiv: 2609.28210
Cross-list: hep-th
Comments: 241 pages, 52 figures. English translation of a book published in Japanese by Saiensu-sha (SGC Library) in September 2015. Some typos and errors in the original have been corrected
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT); Quantum Algebra (math.QA)

This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting -action on 3d theories, 3d supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat -connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.

Official arXiv abstract page

The Thursday, 24 September 2026 arXiv math.GT new-submissions listing contains 11 eligible papers: 3 new submissions and 8 cross submissions. Replacement submissions are excluded.

New submissions

Geometric ideal triangulations of hyperbolic 3-manifolds

Authors: Huabin Ge
arXiv: 2609.27635
Comments: 20 pages, 3 figures. Comments are welcome
Subjects: Geometric Topology (math.GT)

We resolve the longstanding geometric ideal triangulation conjecture arising from Thurston's theory of hyperbolic 3-manifolds: every finite-volume cusped hyperbolic 3-manifold admits a finite geometric ideal triangulation realizing its complete hyperbolic structure.

Official arXiv abstract page

Counting Invariants for Torus Quotients under FPF Group Actions

Authors: Xue Zhang
arXiv: 2609.27736
Comments: 19 pages
Subjects: Geometric Topology (math.GT)

Let $G$ be a finite group admitting a rational fixed-point-free (FPF) representation $V$ , with a $G$ -invariant full-rank lattice $L \subset V$ . We study the induced $G$ -action on the complex torus $V_{C} / L$ and the quotient orbifold $V_{C} / (G ⋉ L)$ . We prove that every FPF group admits a unique irreducible rational FPF representation. We derive closed-form formulas for three core geometric invariants: the number of non-free points of the torus action, the number of singular points of the quotient orbifold, and its orbifold Euler characteristic.

Official arXiv abstract page

Finiteness of closed arithmetic hyperbolic surface bundles

Authors: Tam Cheetham-West; Homin Lee; Nicholas Miller
arXiv: 2609.28188
Comments: 23 pages, 3 figures. Comments welcome!
Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

For each $g \geq 2$ , we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$ -manifolds with genus $g$ fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL $_{2} (C)$ whose image is contained in the fundamental group of some arithmetic hyperbolic $3$ -manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.

Official arXiv abstract page

Cross submissions

Algebraically primitive Teichmüller curves in Ω\mathcal M_g(2g-2)^{\mathrm{hyp}}

Authors: Myeongjae Lee
arXiv: 2609.26903
Cross-list: math.DS
Comments: 17 pages, comments are welcome
Subjects: Dynamical Systems (math.DS); Algebraic Geometry (math.AG); Geometric Topology (math.GT)

We give a complete classification of algebraically primitive Teichmüller curves in $Ω M_{g} (2 g - 2)^{hyp}$ for $g \geq 5$ . They are precisely the curves generated by the double regular $(2 g + 1)$ -gon when $2 g + 1$ is prime, and by the regular $4 g$ -gon when $g$ is a power of two.

Official arXiv abstract page

Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces

Authors: Junehyuk Jung; Insung Park; Peter Zenz
arXiv: 2609.27071
Cross-list: math.DS
Comments: 66 pages, 5 figures
Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG); Geometric Topology (math.GT); Spectral Theory (math.SP)

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $⟨ ϕ, φ^{2} ⟩$ uniformly in $t_{ϕ} \to \infty$ and $t_{φ} \to \infty$ , where $t_{ϕ}$ and $t_{φ}$ are the eigenparameters corresponding to Maass forms $ϕ$ and $φ$ on $X$ .

Official arXiv abstract page

Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

Authors: Krishnendu Gongopadhyay; Neelanjan Mondal
arXiv: 2609.27125
Cross-list: math.DS
Comments: Preliminary version
Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT)

Let $Γ < SL (d, R)$ be a Zariski-dense Borel-Anosov subgroup and let $φ$ be positive on its limit cone. For every $χ \in Hom (Γ, R)$ , we construct twisted $(φ, χ)$ -conformal densities and prove their uniqueness, atomlessness, and ergodicity. For every normal subgroup $Γ_{0} ⊲ Γ$ with nilpotent quotient, we classify the ergodic $φ$ -conformal densities of $Γ_{0}$ in terms of characters of $Γ/ Γ_{0}$ .

Official arXiv abstract page

Finite quotients of spherical Artin groups

Authors: Sam Hughes; Thomas Ng; Kaitlin Ragosta; Nancy Scherich; Yvon Verberne
arXiv: 2609.27171
Cross-list: math.GR
Comments: This is a preliminary preprint which we intend to update in the near future
Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We show the smallest non-abelian quotients of spherical and affine Artin groups are isomorphic to the smallest non-abelian quotients of the corresponding Coxeter groups. We deduce irreducible spherical Artin groups are determined by their finite quotient groups.

Official arXiv abstract page

Bounded cohomology, Codimension two submanifolds and Pontryagin-Thom constructions

Authors: Thorben Kastenholz
arXiv: 2609.27476
Cross-list: math.AT
Comments: This paper represents a rework of the erroneous paper 2503.22511. It represents the attempts of the author to repair and further develop the approach of 2503.22511
Subjects: Algebraic Topology (math.AT); Group Theory (math.GR); Geometric Topology (math.GT)

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

Official arXiv abstract page

Modular integrals for individual modular knots

Authors: Toshiki Matsusaka
arXiv: 2609.27574
Cross-list: math.NT
Comments: 12 pages
Subjects: Number Theory (math.NT); Geometric Topology (math.GT)

To each modular knot, we attach a weight $2$ modular integral for $SL_{2} (Z)$ whose homogenized cycle integrals recover its linking numbers with other modular knots in $S^{3}$ . This construction answers Choie-Zagier's question of explicitly constructing modular integrals with prescribed rational period functions in weight $2$ . Our formula applies to individual modular knots without symmetrization, thereby realizing an approach discussed but not pursued by Duke-Imamoglu-Tóth. This gives an answer to Ghys' question on pairwise linking numbers in terms of modular integrals, as sought by Simon.

Official arXiv abstract page

Milnor's inequality and circular elastic knots

Authors: Tatsuya Miura
arXiv: 2609.27643
Cross-list: math.DG
Comments: 14 pages
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Geometric Topology (math.GT)

We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the $C^{1}$ -closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter--von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti--Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach--Reiter--von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.

Official arXiv abstract page

Geometries of Quantum Field Theories

Authors: Masahito Yamazaki
arXiv: 2609.28210
Cross-list: hep-th
Comments: 241 pages, 52 figures. English translation of a book published in Japanese by Saiensu-sha (SGC Library) in September 2015. Some typos and errors in the original have been corrected
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT); Quantum Algebra (math.QA)

This is an English translation of a book originally published in Japanese by Saiensu-sha in 2015. The book is an introduction to the 3d-3d correspondence, the relation between 3d $N = 2$ supersymmetric gauge theories and the geometry of 3-manifolds, which arises from the compactification of the 6d $N = (2, 0)$ theory. The presentation is bottom-up rather than top-down: instead of starting from the 6d theory, we begin by asking what a quantum field theory is, and let the geometry emerge on its own. Along the way we discuss renormalization and low-energy effective theories, gauging as an operation which glues field theories together, the resulting $S p (2 n, Z)$ -action on 3d theories, 3d $N = 2$ supersymmetric theories and their dualities, the squashed three-sphere partition function and supersymmetric localization, duality domain walls of 4d $N = 2$ theories, quantum Teichmuller theory, complex Chern-Simons theory and the quantization of the moduli space of flat $S L (2, C)$ -connections on 3-manifolds, with quantum dilogarithm functions as a recurring thread. The emphasis throughout is on geometric and algebraic structures which are invisible in a single field theory, and appear only in the theory space of quantum field theories. Appendices summarize supersymmetry in various dimensions, classical and quantum dilogarithm functions, and cluster algebras. Exercises with difficulty ratings are included in each chapter.

Official arXiv abstract page

math.GT daily digest: 11 submissions for 24 September 2026

New submissions

Geometric ideal triangulations of hyperbolic 3-manifolds

Counting Invariants for Torus Quotients under FPF Group Actions

Finiteness of closed arithmetic hyperbolic surface bundles

Cross submissions

Algebraically primitive Teichmüller curves in Ω\mathcal M_g(2g-2)^{\mathrm{hyp}}

Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces

Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

Finite quotients of spherical Artin groups

Bounded cohomology, Codimension two submanifolds and Pontryagin-Thom constructions

Modular integrals for individual modular knots

Milnor's inequality and circular elastic knots

Geometries of Quantum Field Theories

Related content

math.GT daily digest: 11 submissions for 24 September 2026

New submissions

Geometric ideal triangulations of hyperbolic 3-manifolds

Counting Invariants for Torus Quotients under FPF Group Actions

Finiteness of closed arithmetic hyperbolic surface bundles

Cross submissions

Algebraically primitive Teichmüller curves in Ω\mathcal M_g(2g-2)^{\mathrm{hyp}}

Sharp effective equidistribution of closed geodesics in compact hyperbolic surfaces

Twisted Patterson-Sullivan densities on nilpotent covers for Anosov subgroups

Finite quotients of spherical Artin groups

Bounded cohomology, Codimension two submanifolds and Pontryagin-Thom constructions

Modular integrals for individual modular knots

Milnor's inequality and circular elastic knots

Geometries of Quantum Field Theories

Related content