math.GT daily digest: 14 submissions for 23 September 2026

New submissions

Nonabelian Torus Exteriors in and Torus Replacement in Four-Manifolds

• Authors: Anar Akhmedov; Azer Akhmedov
• arXiv: 2609.25080
• Comments: 29 pages
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Group Theory (math.GR)

We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kanenobu-Kazama, with explicitly controlled fundamental groups and peripheral systems, we construct smooth homotopy 4-spheres and manifolds homeomorphic to S^2 x S^2 and #_3(S^2 x S^2). We also study direct gluings of two torus exteriors. A swap gluing of two one-null exteriors kills both meridians and gives a simply connected manifold with Euler characteristic $4$ and signature $0$, while a meridian-preserving gluing of doubly-null exteriors retains the complement groups as an amalgam over the meridian subgroup. Twisted peripheral gluings give cyclic and explicit finite nonabelian fundamental groups. We also consider the full Kanenobu-Kazama $1$-handle family and Litherland tori with full-rank peripheral subgroup; the latter have no null primitive boundary slope and lead naturally to amalgamated-product fundamental groups.

Quantitative asphericalization and Gromov's linear bordism problem

• Authors: Geunho Lim
• arXiv: 2609.25116
• Comments: 31 pages
• Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Metric Geometry (math.MG)

We prove a PL bordism version of Gromov's linearity conjecture over a broad class of asphericalization groups for manifolds endowed with faithful representations in all dimensions. Since every group embeds into an asphericalization group, this bordism version of the conjecture holds after allowing the target group to be enlarged. Our results apply in both the oriented and unoriented settings. In the PL category, no bounded local geometry is required. As an application, we obtain improved linear bounds for the Cheeger--Gromov $\rho$-invariants of PL -manifolds associated with their universal covers, replacing the previous superfactorial growth in dimension by factorial growth. The proofs are based on a quantitative asphericalization method that provides explicit control of the complexity of the resulting bordisms.

Product subcomplexes and intersection complexes of weakly special square complexes

• Authors: Sangrok Oh
• arXiv: 2609.25140
• Comments: 24 pages, 4 figures. Revisits and corrects the constructions of product subcomplexes and intersection complexes in arXiv:2009.03865
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We define product subcomplexes and intersection complexes for compact nonpositively curved square complexes and CAT(0) square complexes. Product subcomplexes are defined as equivalence classes of local isometries from products of graphs with embedded coordinate fibers. Using their factorizations, we give a common definition of the intersection complex, extending the constructions in S. Oh's paper "Quasi-isometry invariants of weakly special square complexes" (Topology and its Applications 307 (2022)). Cubical automorphisms induce actions on these complexes. We define morphisms by comparing compatible elevations of core-map labels and their factorizations along face chains. For a compact two-sided weakly special square complex, we prove that its intersection complex is canonically isomorphic to the deck quotient of the intersection complex of its universal cover. Under a simplicity hypothesis, this recovers the image-defined reduced intersection complex in that paper, and product-base groups are the full simplex stabilizers. We also prove that quasi-isometries between universal covers of compact two-sided weakly special square complexes induce isomorphisms of their intersection complexes in this sense, and give finite counterexamples explaining why images alone do not determine the quotient in general.

4-manifold topology and collision rank of groups

• Authors: Vyacheslav Krushkal
• arXiv: 2609.25198
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

We introduce collision rank and show that groups of finite collision rank are good in the sense of topological $4$-manifold theory. Finitely generated groups of collision rank two are shown to exist outside the class of subexponentially amenable groups. Such examples are given by commutator subgroups of topological full groups of Sturmian systems, and include continuum many pairwise nonisomorphic infinite simple amenable groups.

Cannon-Thurston maps and measures

• Authors: Ursula Hamenstaedt
• arXiv: 2609.25378
• Comments: 9 pages
• Subjects: Geometric Topology (math.GT)

We give a short proof of the following version of a recent result of Gadre, Maher, Pfaff and Uyanki: Consider a closed hyperbolic 3-manifold M which fibers over the circle, with fiber a closed surface S. The measure class on the ideal boundary of hyperbolic 3-space defined by any Gibbs equilibrium state on the unit tangent bundle of M is singular with respect to the push-forward by a Cannon-Thurston map of a measure class on the circle defined by a Gibbs equilibrium state on the unit tangent bundle of S.

Limiting laminations and boundary weights for infinite-type mapping classes

• Authors: Carolyn Abbott; Phuong Pham
• arXiv: 2609.25435
• Subjects: Geometric Topology (math.GT)

For every infinite-type surface with an isolated puncture that admits a shift map and every natural number $n$, we study an intrinsically infinite-type mapping class and a simple closed curve whose orbit under splits into $n$ subsequences, each of which converges in the coarse Chabauty topology to a geodesic lamination on the surface. These laminations are cyclically permuted by , and their union is a --invariant geodesic lamination. We further show the attracting limit point of in the boundary of the relative arc graph has weight $n$. In particular, intrinsically infinite-type loxodromic isometries of the relative arc graph realize every finite weight.

A new conformal gauge on the visual boundary of a hyperbolic group

• Authors: Manisha Garg
• arXiv: 2609.25468
• Comments: 46 pages, 1 Figure
• Subjects: Geometric Topology (math.GT); Metric Geometry (math.MG)

We study a logarithmic re-gauging of the visual boundary, arising from the logarithmic metric on the sublinear Morse boundary introduced by Garg-Jana-Qing. For every proper geodesic hyperbolic space $X$, we prove that . As a consequence, under the natural identification of boundaries, is bi-Lipschitz equivalent to the polynomial Floyd boundary metric associated with the radial density . Every quasi-isometry between proper geodesic hyperbolic spaces induces a bi-Lipschitz map between their logarithmic boundaries with a fixed admissible exponent.

For every non-elementary hyperbolic group, the logarithmic boundary is nondoubling and has infinite Hausdorff and Assouad dimensions. Nevertheless, the logarithmic re-gauging preserves Nagata (capacity) dimension and continues to recover the asymptotic dimension of the group. It also detects one-endedness through linear connectedness (bounded turning) and preserves uniform perfectness. Thus the logarithmic and visual metrics are not quasisymmetrically equivalent, although several coarse-geometric features remain visible in the logarithmic boundary.

Plane separating continua inscribe rectangles

• Authors: Ulises Morales-Fuentes; Cristina Villanueva-Segovia
• arXiv: 2609.25574
• Comments: 13 pages
• Subjects: Geometric Topology (math.GT)

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

Every Link Has Infinitely Many Explicit Generalised T-Link Presentations

• Authors: Thiago de Paiva; Yi Liu
• arXiv: 2609.25683
• Comments: 11 pages
• Subjects: Geometric Topology (math.GT)

Generalised $T$-links provide a simple description of all links in as closures of products of standard twisting blocks, parametrised by finite sequences of integers. We prove that every link admits infinitely many pairwise distinct generalised $T$-link presentations. Starting from any such presentation, we give explicit parameter transformations that preserve the represented link and generate families of pairwise distinct presentations depending on arbitrarily many independent integer parameters.

On the Sp-structure of the torsion of the Lie algebra of homology cylinders

• Authors: Quentin Faes; Gwenael Massuyeau; Masatoshi Sato
• arXiv: 2609.26409
• Comments: 76 pages
• Subjects: Geometric Topology (math.GT)

Let $\Sigma$ be a compact oriented surface. We investigate the graded Lie algebra of homology cylinders over $\Sigma$, as introduced by M. Goussarov and K. Habiro. To analyze its torsion, we refine a strategy initiated by Y. Nozaki, M. Suzuki, and the third author, which relies on the reduction modulo $1$ of the LMO functor and on clasper calculus. Specifically, we develop general tools for studying, under the standard action of the symplectic group, the Sp-module structure of the torsion in the odd-degree component of this Lie algebra. We show that this torsion part surjects onto an $2$-torsion Sp-module, which is explicitly described in terms of Jacobi diagrams. As an application, we provide an intrinsic description of the Sp-module given by the degree-three component of the Lie algebra of homology cylinders. A further motivation is to understand torsion phenomena in the associated graded of the lower central series of the Torelli group of $\Sigma$: in this direction, we exhibit an explicit Sp-module onto which the torsion of the Torelli Lie algebra surjects in degree three.

Irreducible exotic 's with free involutions

• Authors: R. İnanç Baykur; András I. Stipsicz; Zoltán Szabó
• arXiv: 2609.26450
• Subjects: Geometric Topology (math.GT)

We construct infinitely many pairwise non-diffeomorphic irreducible smooth four-manifolds homeomorphic to , each admitting a free, orientation-preserving involution. Their quotients give infinitely many irreducible definite four-manifolds with fundamental group \mathbb {Z}}/2 {\mathbb {Z}! and , filling the corresponding gap in the geography of such examples. The construction of infinitely many distinct examples relies on a novel equivariant torus surgery.

On Sections of Quotient maps by Isometries

• Authors: Suyoung Choi; Tao Gong
• arXiv: 2609.26452
• Comments: 5 pages. Comments are welcome
• Subjects: Geometric Topology (math.GT)

Suppose that a discrete group $G$ acts smoothly, properly, and effectively on a connected smooth manifold $M$, and let be the quotient map. We prove that if $q$ admits a continuous section, then $G$ is generated by reflections. We also establish converse results for dissecting reflections and give applications to Euclidean and lattice actions.

Verified Length Spectrum and Margulis number for Hyperbolic 3-Manifolds

• Authors: Matthias Goerner; Robert C. Haraway III; Neil R. Hoffman; Maria Trnkova
• arXiv: 2609.26663
• Comments: 53 Pages, 8 Figures, 4 Algorithms
• Subjects: Geometric Topology (math.GT)

We present a new algorithm to compute a stream for the length spectrum of an oriented, finite-volume hyperbolic 3-manifold $M$. The advantage over the existing algorithm by Hodgson-Weeks is that our algorithm does not require finding a Dirichlet domain and thus can use interval arithmetic to verify the result. Our algorithm is also significantly faster in some cases. We also present a new algorithm to compute the optimal Margulis number of $M$. The authors have made these algorithms available in SnapPy.

Cross submissions

Smallest quotients and profinite rigidity of irreducible spherical type Artin groups

• Authors: Federica Gavazzi; Igor Haladjian; Luis Paris
• arXiv: 2609.25940
• Cross-list: math.GR
• Comments: 38 pages, 1 figure. Comments welcome!
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)

We prove that irreducible spherical type Artin groups are profinitely rigid within the class of all spherical type Artin groups. As part of the proof, building on ideas of Kolay for the $n$-strand braid group, we compute the smallest non-abelian quotient of every spherical type Artin group. In particular, given an irreducible spherical type Artin group, its smallest non-abelian quotient is always that of the corresponding Coxeter group, except for dihedral Artin groups of weight divisible by $4$, which all have as a non-abelian quotient. Parts of the proof also rely on the computation of the cohomological dimension of the profinite completion of certain Artin groups.