math.GT daily digest: 14 new submissions for 29 July 2026

New submissions

The Birman--Craggs--Johnson homomorphism and the handlebody Torelli group

• Authors: Annie Holden
• arXiv: 2607.25191
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
• Comments: 23 pages, 5 figures

We use the Birman--Craggs--Johnson (BCJ) homomorphism to study the intersection of the handlebody group with the Torelli group and with the Johnson kernel. For genus $\ge 3$, we determine the images of the BCJ homomorphism restricted to these subgroups. We then explicitly compute cup products of pairs of cohomology classes in detected by the BCJ homomorphism for genus $\ge 4$. For the handlebody Johnson kernel, our results lift to integral coefficients.

The Farey tree and fibrations of the Whitehead link complement

• Authors: Tali Pinsky
• arXiv: 2607.25194
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)
• Comments: 17 pages, 18 figures

We show a curious connection between an example of dynamical forcing, where an existence of one periodic orbit for a dynamical system forces the existence of other periodic orbits, and the existence of different fibrations of a 3 dimensional manifold. Specifically, we establish a one to one correspondence between fibrations of the Whitehead link complement and the set of \`simple orbit pairs' on the torus. The set of orbits is equipped with a complete order coming from a dynamical forcing relation, that corresponds to the Farey tree order of rational numbers. This corresponds to the order of the rational point on a fibered cone of the Whitehead link. For the proof we obtain an explicit description of all monodromies of the different fibrations of the Whitehead link complement.

Insensitivity of Khovanov homology under rim surgery

• Authors: Qiuyu Ren; Fang-Rong Zhan
• arXiv: 2607.25302
• Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
• Comments: 8 pages, 2 figures

We show that rim surgery on a smooth, oriented, properly embedded surface in does not change the map on Khovanov homology induced by the surface, answering a question raised by Hayden and Sundber. More generally, the same insensitivity holds for a class of local surgery operations that we call local annulus replacements. Combined with the work of Hayden--Sundberg, this result yields exotic pairs of surfaces in that cannot be related by any sequence of local annulus replacements. We give two proofs, both making essential use of Khovanov skein lasagna modules.

On the Asymptotics of Convex Core Volumes of Once-Punctured Torus Groups

• Authors: Hidetoshi Masai; Ryo Matsuda
• arXiv: 2607.25319
• Subjects: Geometric Topology (math.GT)
• Comments: 31 pages, 4 figures

We study the asymptotic behavior of the convex core volume for a sequence of quasi-Fuchsian manifolds associated with a pseudo-Anosov mapping class $\psi$ on a once-punctured torus. We prove that the volume of the convex core differs from $2n$ times the volume of the mapping torus of $\psi$ by at most a uniformly bounded constant.

On a twisted Singer conjecture

• Authors: Jacopo G. Chen
• arXiv: 2607.25615
• Subjects: Geometric Topology (math.GT)
• Comments: 17 pages

According to the Singer conjecture, the -Betti numbers of a closed aspherical manifold are expected to vanish outside the middle dimension. In this paper, we study a natural analog of the Singer conjecture concerning the vanishing of twisted -Betti numbers outside the lower middle dimension. These invariants can be thought of as -Betti numbers of an infinite cyclic cover, depending on a first cohomology class. This is related to a previous conjecture by the author, which connects the twisted -Euler characteristic to the Thurston norm, and which we prove for a class of closed arithmetic hyperbolic manifolds.

Spherical orthogonal ring patterns on surfaces and modified combinatorial total geodesic curvatures

• Authors: Zhiwen Xiong; Xu Xu
• arXiv: 2607.25705
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

Orthogonal ring patterns are natural generalizations of circle patterns. Bobenko-Hoffmann-Rörig and Bobenko established the variational principles of the classical combinatorial curvature for the Euclidean, hyperbolic and spherical orthogonal ring patterns. Bobenko-Hoffmann-Rörig's work and Bobenko's work imply the rigidity of Euclidean and hyperbolic orthogonal ring patterns on closed surfaces, while the rigidity of spherical orthogonal ring patterns on closed surfaces is not known. In this paper, we study the spherical orthogonal ring patterns on closed surfaces with cellular decompositions satisfying certain necessary conditions. Using a modification of the combinatorial total geodesic curvature introduced by Nie in \cite{Nie}, we prove the rigidity of spherical orthogonal ring patterns on closed surfaces by variational principles.

Algebraic concordance groups for links

• Authors: Gaëtan Simian
• arXiv: 2607.25931
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT)
• Comments: 31 pages, 10 figures

We use families of generalized Seifert matrices to define algebraic concordance groups for links with non trivial Alexander polynomial. We then use these groups to recover the invariance by concordance of the signature, the Fox-Milnor condition for concordant links and the invariance by concordance of the Witt class of the Blanchfield pairing for links. Finally, we use these invariants to investigate the isomorphism types of these groups.

Algebraic concordance of links

• Authors: David Cimasoni; Anthony Conway; Gaetan Simian
• arXiv: 2607.25972
• Subjects: Geometric Topology (math.GT)
• Comments: 61 pages, 15 figures

Algebraic concordance of knots can be understood from the perspective of Seifert matrices, Blanchfield forms, and homology surgery. We initiate a systematic study of algebraic concordance for links from each of these viewpoints. The present article is concerned with algebraic concordance from the perspective of homology surgery and Blanchfield forms, whereas a companion article by the third named author focuses on C-complexes and generalised Seifert matrices. The outcome of the present work consists of two obstructions to $\mu$-component links being concordant. The first obstruction, called the homology surgery invariant, takes values in the Witt group of hermitian forms over the field of fractions $Q$ of \mathbb{Z}\\[\mathbb{Z}^\mu\\]!. The second obtruction, called the Blanchfield invariant, takes values in a Witt group of Q/\mathbb{Z}\\[\mathbb{Z}^\mu\\]!-valued hermitian linking forms. For , we describe these invariants in terms of generalised Seifert matrices.

Cross submissions

Topology and dynamics of unimodular random hyperbolic manifolds

• Authors: Ilya Gekhtman; Nir Lazarovich; Arie Levit; Asaf Nachmias
• arXiv: 2607.25065
• Cross-list from: math.DS
• Subjects: Dynamical Systems (math.DS); Geometric Topology (math.GT); Probability (math.PR)
• Comments: 43 pages. 1 figure. glossary of notations

We investigate the relationship between the space of ends of a unimodular random hyperbolic manifold and the dynamics of its geodesic flow. We show that having two ends of infinite volume implies recurrence, while having infinitely many such ends implies positive drift and entropy. We also provide a transience criterion which applies to deterministic hyperbolic manifolds. Our method relies on studying Delaunay graphs over point processes and on the analytic notion of capacity.

Sublinear projection tracking in acylindrically hyperbolic groups

• Authors: Lihuang Ding; Wenyuan Yang
• arXiv: 2607.25555
• Cross-list from: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 41 pages, 4 figures

We study projection phenomena in word metrics of finitely generated acylindrically hyperbolic groups. For a loxodromic WPD element acting on a hyperbolic space, we prove that shortest projection in the word metric to the corresponding cyclic subgroup sublinearly tracks the pullback of shortest projection to its axis in the hyperbolic space. As applications, we obtain effective upper bounds for growth functions and construct proper quotients whose growth rates converge to that of the original group. We further prove a growth--cogrowth inequality for confined subgroups in both acylindrically hyperbolic groups and Morse local-to-global groups with Morse elements.

Random Trees in Hyperbolic FPP

• Authors: Riddhipratim Basu; Mahan Mj
• arXiv: 2607.25567
• Cross-list from: math.GR
• Subjects: Probability (math.PR); Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 28 pages, 2 figures. Submitted to the Proceedings of the International Colloquium on randomness, geometry and dynamics, 2024

We start by surveying known properties of first passage percolation (FPP) geodesics on a Gromov-hyperbolic group $G$. Due to a coalescence phenomenon established in earlier work of the authors, a random tree consisting of infinite random geodesics to a point $\xi$ on the Gromov boundary $\partial G$ emerges naturally. These trees have a rich random geometry that we explore further in this paper.

Growth gaps and exponential genericity in acylindrically hyperbolic groups

• Authors: Lihuang Ding; Wenyuan Yang
• arXiv: 2607.25578
• Cross-list from: math.GR
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 46 pages

We prove that, for every finite generating set of an acylindrically hyperbolic group, the set of non-WPD elements has strictly smaller exponential growth rate. Equivalently, WPD elements are exponentially generic. As applications, we prove growth tightness and cogrowth tightness for acylindrically hyperbolic groups.

Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

• Authors: Yuerong Bian; Xu Xu
• arXiv: 2607.25691
• Cross-list from: math.DG
• Subjects: Differential Geometry (math.DG); Geometric Topology (math.GT)

We study the combinatorial Yamabe flow on infinitely triangulated surfaces with piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and $ϵ$-uniformly nondegenerate initial metric, we first establish the short-time existence of smooth solutions to the combinatorial Yamabe flow. Under the additional $ϵ$-uniformly Delaunay condition on the initial metric, we further obtain the short-time uniqueness of solutions to the flow. To address the potential degeneration of triangles along the evolution, we introduce an extended flow with generalized curvature, and establish the global existence of solutions to the extended flow. Furthermore, under uniformly bounded vertex degrees and some integrability condition, we establish the uniqueness of solutions to this extended flow, which follows from the stability property of the solutions. These results provide a well-posedness theory for both the hyperbolic combinatorial Yamabe flow (locally in time) and its extended flow (globally in time) on infinitely triangulated surfaces.

Contact Surgery Numbers of the 3-torus

• Authors: Prerak Deep; Monika Yadav
• arXiv: 2607.25772
• Cross-list from: math.SG
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)
• Comments: 22 Pages, 3 figures, 1 Table. Comments are welcome

We study contact surgery numbers for contact structures on the 3-torus. We show that all contact structures obtained via contact surgery along a Legendrian Borromean ring are overtwisted, and that this construction yields infinitely many pairwise non-contactomorphic contact structures on . We prove an obstruction on the maximal Thurston-Bennequin invariant of 3-component links to produce by Dehn surgery. On a side note, using Legendrian surgery and the ruling invariant, we show that the total symplectic homology of Stein fillings of is non-zero.