math.GT daily digest: 9 new submissions for 16 July 2026

New submissions

Inscribed squares of level sets of functions on the sphere

• Authors: V.A. Vassiliev
• arXiv: 2607.13052
• Subjects: Geometric Topology (math.GT); Classical Analysis and ODEs (math.CA)

Every continuous function takes the same value on the vertices of a square inscribed into a great circle of . Keywords: level set, configuration space, Stiefel--Whitney class, inscribed squares, Borsuk-Ulam type theorems

Simultaneous Laminations

• Authors: Isaac Broudy
• arXiv: 2607.13676
• Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS)

Calegari introduced the laminations associated to a universal circle. We study the laminations for pseudo-Anosov orbit space universal circles of taut foliations, on atoroidal three-manifolds. We prove that and are completely determined by the stable and unstable prelaminations on the boundary of the orbit space. Then, using a result of Barthelmé, Bonatti, and Mann, we prove that for any action coming from an orbit space of a pseudo-Anosov flow, there is a finite collection of lamination pairs such that lies in this collection for any minimal orbit space universal circle whose action is conjugate to .

Filling 1-cycles, 2-cycle complexity, and torsion growth

• Authors: Cameron Gates Rudd
• arXiv: 2607.13736
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)

For 2-complexes with trivial first Betti number, we study quantitative connections between filling inequalities for 1-cycles, the complexity of the second homology, and the size of the torsion part of the first homology. For 2-complexes whose fundamental groups have property with respect to all finite index normal subgroups, we prove a geometric lower bound on the logarithm of the size of the first homology of finite covers.

Combinations and chromatography of paths of Kleinian groups

• Authors: Matthew Zevenbergen
• arXiv: 2607.13950
• Subjects: Geometric Topology (math.GT)
• Comments: 31 pages, 3 figures. Subsection 4.2 was split from v1 of arXiv:2410.15537 and heavily revised

We study continuous paths in the Chabauty topology on the set of torsion free discrete subgroups of the isometry group of -dimensional hyperbolic space. We prove a combination theorem for paths in , which allows us to construct an exotic path of discrete subgroups along which no two subgroups are isomorphic. We also introduce a technique we refer to as "chromatography" to prove a decomposition theorem that characterizes paths of convex cocompact groups in .

Braid closure union braid axis is ribbon concordance minimal

• Authors: Benjamin Daniels
• arXiv: 2607.14030
• Subjects: Geometric Topology (math.GT)
• Comments: 7 pages, comments welcome!

We show that a ribbon concordance minimal fibered knot in can generate ribbon concordance minimal links by the addition of any braid closure in . As a corollary, we show that any link in may be made ribbon concordance minimal by adding a single unknot linked with . Our proofs use link Floer homology together with classical techniques.

Cross submissions

Braid groups and Burnside groups

• Cross-list from: math.GR
• Authors: Ethan Dlugie
• arXiv: 2607.13316
• Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 10 pages, 2 figures

There exists an exceptional quotient of braid groups that is related to many interesting constructions in algebra, topology, and geometry. This quotient map also descends to a quotient of "truncated" braid groups , which have an added torsion relation on their half twist generators. In this article, we find a presentation for the kernel of this truncated quotient map that takes the form of what we deem a "primitive" Burnside group. We give a few finiteness results on these primitive Burnside groups. Our methods are purely group theoretic, but we comment on an interpretation involving Lefschetz fibrations at the end.

Representability of systems of proportionally modular numerical semigroups

• Cross-list from: math.NT
• Authors: Zsolt Baja; Tamás László; Zsuzsa Nagy
• arXiv: 2607.13619
• Subjects: Number Theory (math.NT); Geometric Topology (math.GT)
• Comments: 12 pages

In this short note we prove that every system of proportionally modular numerical semigroups is representable by a canonical equivariant resolution of a weighted homogeneous surface singularity with rational homology sphere link. The construction starts from the quotient descriptions of proportionally modular numerical semigroups by two-generator numerical semigroups, realizes each quotient by a two-legged canonical equivariant resolution graph, and then glues these graphs with suitable multiplicities.

Regularity of Manhattan manifolds and exact dimensionality for relatively Anosov groups

• Cross-list from: math.GR
• Authors: Eduardo Reyes; Tianqi Wang
• arXiv: 2607.13730
• Subjects: Group Theory (math.GR); Dynamical Systems (math.DS); Geometric Topology (math.GT)
• Comments: 30 pages

We establish several results about Patterson--Sullivan measures for relatively Anosov groups. First, we prove that these measures are exact dimensional with respect to visual metrics induced by Gromov models in the Groves--Manning quasi-isometry class. Under the additional assumption that the group is relatively Morse, we show that the associated scalar Cartan metric is Gromov hyperbolic and that the corresponding boundary premetric is a visual metric to which the exact-dimensionality theorem applies. Second, we prove that their Manhattan manifolds are -regular, from which we deduce that the growth indicator is -regular and strictly concave on the interior of the limit cone. This extends the case of Anosov representations by Kim--Oh--Wang. Our methods are dynamical, and we exploit the fact due to Kim--Oh and Blayac--Canary--Zhu--Zimmer that Bowen--Margulis--Sullivan measures for relatively Anosov groups are finite and mixing.

Strict concavity of the growth indicator function for relatively Anosov groups

• Cross-list from: math.DG
• Authors: Dongryul M. Kim; Hee Oh; Andrew Zimmer
• arXiv: 2607.13760
• Subjects: Differential Geometry (math.DG); Dynamical Systems (math.DS); Group Theory (math.GR); Geometric Topology (math.GT)
• Comments: 31 pages. Comments welcome!

Let be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function records the directional exponential growth of the Cartan projections of elements of in the positive Weyl chamber . We prove that if is a non-elementary relatively Borel Anosov group, then is strictly concave on non-collinear directions. We prove this by establishing the -smoothness of the Manhattan hypersurface, defined as the unit level set of the critical-exponent map . More generally, for a non-elementary -transverse group, we prove local -regularity near every point of the -Manhattan hypersurface that is positive on the -limit cone and has a critical gap at infinity. In particular, the -Manhattan hypersurface is globally for relatively -Anosov groups, and their -growth indicator functions are strictly concave on non-collinear directions.