math.GT daily digest: 8 new submissions for 20 August 2026

New submissions

Discrete embeddings of hyperbolic groups with Pontryagin-surface boundaries

• Authors: Jiming Ma, Junseo Yoon, Fangting Zheng
• arXiv: 2608.18163
• Subjects: Geometric Topology (math.GT)
• Comments: 40 pages

Let be the quotient of the closed disk obtained by identifying boundary points under rotation through angle $2\pi /p$. For every , we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to that admits a discrete, faithful, convex cocompact reflection representation into Isom, whose limit set is homeomorphic to the index-$p$ Pontryagin surface . Dimension five is optimal, since does not embed in . For , the constructions are analytic and yield cyclically symmetric infinite families.

Minimal Filling pair of non orientable surfaces

• Authors: Debattam Das, Souvik Pal, Bidyut Sanki
• arXiv: 2608.18848
• Subjects: Geometric Topology (math.GT)
• Comments: 23 pages, 9 figures

For , let denote the non-orientable surface of genus $g$. In this article, we establish the existence of filling pairs on that intersect minimally by construction using the theory of fat graphs. The mapping class group acts on the set of all such filling pairs. We count -orbits of this action by providing both lower and upper bounds. Furthermore, we show that both bounds grow super-exponentially with $g$ using graph cohomology. Also, we investigate the lengths of minimally intersecting filling pairs on hyperbolic non-orientable surfaces $X$ in moduli space of . We define a function , where for , the function is the shortest total length of a minimally intersecting filling pair on $X$. We determine its minimum and show that the set of minimizers is in bijection with the -orbits of minimally intersecting filling pairs. We further extend to , defined by minimizing the length over all filling pairs, and show that attains the same minimum value as .

Actions on CAT(-1) spaces with critical exponent less than 1

• Authors: Beibei Liu, Shi Wang
• arXiv: 2608.18906
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR); Metric Geometry (math.MG)
• Comments: 16 pages

We show that for a discrete isometry subgroup acting on a proper CAT(-1) space X, if the critical exponent is less than $1$, then the critical exponent equals the Hausdorff dimension of the entire limit set. Consequently, the limit set must be a Cantor set. As an application, we prove that any finitely generated, torsion-free discrete subgroup in Isom(X) with critical exponent less than one must be geometrically finite and free. This answers a question of Kapovich.

Universal braids for elliptic fibrations: from character varieties to Coxeter's factor groups

• Authors: Faye Jackson
• arXiv: 2608.19138
• Subjects: Geometric Topology (math.GT)
• Comments: 30 pages, 6 figures

Let be an elliptic fibration over or with $n$ nodal fibers over . We study the universal liftable braids for $\pi$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B,\Delta )$. When , we show that nontrivial universal braids do not exist by proving a Zariski-density theorem on the -character variety for . When we classify when the subgroup of universal braids has finite index in the braid group , and relate these examples to Coxeter's factor groups of braid groups, which in turn are related to the platonic solids. Finally, we generalize the results derived in the finite-index cases by considering a canonical family of branched covers of the base $B$ associated to any elliptic fibration. The generalization naturally connects the universal braids to the integral Burau representation reduced modulo 3.

Cross submissions

Exceptional eigenvalue density for thin groups

• Cross-list from: math.SP
• Authors: Christopher Lutsko
• arXiv: 2608.18236
• Subjects: Spectral Theory (math.SP); Geometric Topology (math.GT)
• Comments: 7 pages; comments welcome

We prove a limit multiplicity conjecture of Hee Oh for principal congruence covers of geometrically finite hyperbolic manifolds, with an explicit power-saving rate. The rate is governed by the return of Patterson--Sullivan shadows through a fixed compact core, giving a geometric interpretation to the exceptional eigenvalue density. For convex-cocompact groups, a packing argument for enlarged shadows gives a stronger rate.

On Ruling polynomials of Legendrian links

• Cross-list from: math.SG
• Authors: Orsola Capovilla-Searle, Yu Pan
• arXiv: 2608.18255
• Subjects: Symplectic Geometry (math.SG); Geometric Topology (math.GT)
• Comments: 32 pages, 29 figures

The ruling polynomial is a Legendrian invariant that is closely related to the augmentation variety of the Legendrian. We characterize all graded and ungraded ruling polynomials, and construct Legendrian links realizing each possible polynomial. The graded augmentation varieties of the Legendrians we construct all have trivial cluster algebra structures. Finally, we construct Legendrian knots admitting $k$ exact Lagrangian fillings with that are pairwise smoothly non-isotopic for , and

The Brauer category $\mathcal{B}(2)$ has principal graph

• Cross-list from: math.RT
• Authors: Stephen Bigelow
• arXiv: 2608.18328
• Subjects: Representation Theory (math.RT); Geometric Topology (math.GT)
• Comments: 11 pages

We show that the subfactor planar algebra with principal graph is the Brauer planar algebra with bubble constant . The Brauer algebra is similar to the Temperley-Lieb algebra, but with virtual crossings. At , it relates to the category of representations of the orthogonal group $O(2)$. We work over any commutative ring with $1/2$. Its principal graph encodes information about the corresponding monoidal category.

Learning Topological Features of -invariants

• Cross-list from: hep-th
• Authors: Brandon Robinson, Shimal Harichurn, Fabian Ruehle, Sergei Gukov, Rak-Kyeong Seong, Miranda C. N. Cheng
• arXiv: 2608.18570
• Subjects: High Energy Physics - Theory (hep-th); Machine Learning (cs.LG); Geometric Topology (math.GT)
• Comments: 77 pages, 25 figures

Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of -invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the -invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that -invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.