math.GT daily digest: 4 new submissions for 20 July 2026

New submissions

A combinatorial approach to obstructing left-orderability of Dehn fillings

• Authors: Adam Clay; Junyu Lu; Tjay Staruch
• arXiv: 2607.15603
• Subjects: Geometric Topology (math.GT)
• Comments: 21 pages, 1 figure

We introduce the conjugate-slope property of knots, a variant of Nie's Property (D), which can be used to study non-left-orderability of fundamental groups arising from Dehn filling. We show that this property holds for any knot in admitting a diagram whose negative crossings can be "controlled" in a precise combinatorial sense. The conjugate-slope property also places strong restrictions on the kinds of left-orderings that the knot group can support, and implies that the image of the meridian is generalised torsion in the fundamental group of many of the knot's Dehn fillings.

The pants graph as a combinatorial model for the Teichmüller space of a non-orientable surface

• Authors: Michał Stukow; Błażej Szepietowski; Jakub Szmelter-Tomczuk
• arXiv: 2607.15792
• Subjects: Geometric Topology (math.GT)

We study the relation between the pants graph of a non-orientable surface and that of its orientable double cover. Given a non-orientable surface $N$ with orientable double cover , we construct a natural map between pants graphs induced by lifting pants decompositions. We prove that this map defines a quasi-isometric embedding of $\mathrm{Pants}(N)$ into $\mathrm{Pants}(S)$. Using Brock's quasi-isometry between $\mathrm{Pants}(S)$ and Teichmüller space $\mathrm{Teich}(S)$ endowed with the Weil-Petersson metric, together with the identification of the Teichmüller space of $N$ with the fixed-point locus of the deck involution on $\mathrm{Teich}(S)$, we prove that $\mathrm{Pants}(N)$ is quasi-isometric to the Teichmüller space of $N$ equipped with the induced Weil-Petersson metric.

Dirichlet domains and the NEC uniformization of extremal hyperbolic surfaces

• Authors: Ernesto Girondo; Claudia Muñoz
• arXiv: 2607.15882
• Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)
• Comments: 34 pages, 19 figures

Extremal hyperbolic surfaces are finite-type hyperbolic surfaces that contain an embedded disc of the largest possible radius. We introduce a new general approach to these objects, based on the uniformization by NEC groups, obtaining a structural description of the Dirichlet domains associated to extremal disc centers for surfaces with cusps and/or geodesic boundary. We also provide a number of applications of our new description of extremal surfaces, as an effective counting formula for extremal disc configurations, the determination of every such configuration with Euler characteristic -1 and -2, the location of hidden extremal discs in several families of extremal surfaces, and the full description of their automorphism groups.

Profinite rigidity of simple closed curves in surface groups

• Authors: Zhongzi Wang; Xiaoyu Xu
• arXiv: 2607.16147
• Subjects: Geometric Topology (math.GT); Group Theory (math.GR)
• Comments: 24 pages, 2 figures

This paper establishes a new characterization of simple closed curves on a closed orientable surface. Let $\Gamma$ be the fundamental group of a closed orientable surface. We prove that if an element has the same possible images as a given simple closed curve under epimorphisms from $\Gamma$ to every finite group, then $g$ belongs to the $\mathrm{Aut}(\Gamma )$-orbit of $\gamma$, i.e. $g$ is itself a simple closed curve with the same topological type as $\gamma$. Consequently, the set of simple closed curves in $\Gamma$ is closed in the profinite topology of $\Gamma$; and we obtain a new algorithm to decide whether a given element in $\Gamma$ can be represented by a simple closed curve. Proper powers of simple closed curves and the pro-$p$ cases are also discussed.